Reference

Formula Index

A fast lookup for the formulas behind the tools. Search by method, topic, or application, then open the matching calculator when you want the substitution written out.

ConceptTopicFormulaInputsWatch for
Tool route map Vectors and 3D geometry \text{coordinates}\rightarrow\text{curves}\rightarrow\text{complex numbers}\rightarrow\text{vectors} none N/A
Circle with centre and radius Coordinate geometry (x-g)^2+(y-f)^2=r^2 g|f|r r must be non-negative.
Circle centred at the origin Coordinate geometry x^2+y^2=r^2 r r must be non-negative.
General circle equation to centre-radius form Coordinate geometry x^2+y^2-2gx-2fy+c=0 x_coefficient|y_coefficient|constant Radius squared may be negative, meaning no real circle.
Tangent to circle at a point Coordinate geometry xx_1+yy_1=r^2 x1|y1|r Point should lie on the circle.
Classify a second-degree equation with parallel axes Coordinate geometry ax^2+by^2+2gx+2fy+c=0 a|b|g|f|c Degenerate cases need warning, not confident classification.
Ellipse in standard form Coordinate geometry \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 a|b a and b must be positive.
Hyperbola in standard form Coordinate geometry \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 a|b a and b must be positive.
Parabola in standard form Coordinate geometry y^2=4ax a a cannot be zero.
Angle between two lines Coordinate geometry \tan\theta=\frac{m_2-m_1}{1+m_1m_2} m1|m2 Perpendicular when 1+m1*m2=0.
Distance between two points Coordinate geometry d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} x1|y1|x2|y2 None, but identical points give distance 0.
General line form Coordinate geometry ax+by+c=0 a|b|c a and b cannot both be zero; b=0 gives vertical line.
Gradient / slope of a line Coordinate geometry m=\frac{y_2-y_1}{x_2-x_1} x1|y1|x2|y2 Vertical line when x2=x1; slope is undefined.
Line using x- and y-intercepts Coordinate geometry \frac{x}{a}+\frac{y}{b}=1 x_intercept_a|y_intercept_b a or b cannot be zero in this form.
Intersection of two slope-intercept lines Coordinate geometry x=\frac{c_2-c_1}{m_1-m_2},\quad y=\frac{m_1c_2-c_1m_2}{m_1-m_2} m1|c1|m2|c2 Parallel/coincident when m1=m2.
Equation of a line through two points Coordinate geometry y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) x1|y1|x2|y2 Vertical line when x2=x1; output x=x1.
Line through a point with a known gradient Coordinate geometry y-y_1=m(x-x_1) x1|y1|m None for finite m.
Shortest distance from point to line Coordinate geometry d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}} a|b|c|x1|y1 a and b cannot both be zero.
Slope-intercept form Coordinate geometry y=mx+c m|c None for finite m and c.
Regions defined by circle inequalities Inequalities x^2+y^2-r^2 \lessgtr 0 r|inequality_type Boundary included for <= or >=.
Region defined by a linear inequality Inequalities ax+by+c \lessgtr 0 a|b|c|inequality_type a and b cannot both be zero.
Regions defined by parabola inequalities Inequalities y^2-4ax \lessgtr 0 a|inequality_type Boundary included for <= or >=.
Feasible region for simultaneous linear inequalities Inequalities a_1x+b_1y+c_1 \lessgtr 0,\; a_2x+b_2y+c_2 \lessgtr 0,\; ... list_of_inequalities Unbounded, empty, or boundary-only regions.
Curve sketching feature checklist Curve sketching y=f(x) function Depends on parser support; MVP can use preset examples.
Even and odd symmetry Curve sketching f(-x)=f(x) \text{ or } f(-x)=-f(x) function_or_preset Requires parser; can start with manual/preset examples.
Increasing and decreasing intervals from derivative sign Differentiation f'(x)>0 \Rightarrow f \text{ increasing},\quad f'(x)<0 \Rightarrow f \text{ decreasing} derivative_expression_or_critical_points Needs symbolic or preset examples.
Archimedean spiral Coordinate geometry r=a\theta a|theta_min|theta_max Requires angle range; explain radians.
Convert Cartesian coordinates to polar coordinates Coordinate geometry r=\sqrt{x^2+y^2},\quad \theta=\operatorname{atan2}(y,x) x|y Origin has r=0 and angle is convention-dependent.
Convert a polar curve into Cartesian form Coordinate geometry x=r\cos\theta,\quad y=r\sin\theta,\quad r^2=x^2+y^2 polar_equation Not every polar equation converts neatly; some need implicit form or numerical plotting.
Plot a polar curve by sampling theta Coordinate geometry x=r(\theta)\cos\theta,\quad y=r(\theta)\sin\theta r_theta_expression|theta_min|theta_max|step Negative r values plot on the opposite ray; handle with explanation.
Convert polar coordinates to Cartesian coordinates Coordinate geometry x=r\cos\theta,\quad y=r\sin\theta r|theta Angle units must be clear.
Old coordinates from rotated axes coordinates Coordinate geometry x=x'\cos\alpha-y'\sin\alpha,\quad y=x'\sin\alpha+y'\cos\alpha x_prime|y_prime|alpha Angle units must be clear. Positive alpha means counter-clockwise rotation of axes.
Rotated axes coordinates from old coordinates Coordinate geometry x'=x\cos\alpha+y\sin\alpha,\quad y'=-x\sin\alpha+y\cos\alpha x|y|alpha Angle units must be clear. This is the inverse relation of the forward transform.
Chord joining two parameter points on a parabola Polar and parametric curves y(t_1+t_2)=2x+2at_1t_2 a|t1|t2 If t1=t2, the chord becomes the tangent at that parameter.
Cycloid generated by a rolling circle Polar and parametric curves x=a(\theta-\sin\theta),\quad y=a(1-\cos\theta) a|theta_min|theta_max|sample_count Theta should normally be measured in radians. One arch runs from 0 to 2π.
Circle as a parametric curve Polar and parametric curves x=a\cos\theta,\quad y=a\sin\theta a|theta a must be positive; theta may be degrees or radians.
Parametric representation of a curve Polar and parametric curves x=f(t),\quad y=g(t) x_t|y_t|t_min|t_max|step Different parameter values can sometimes produce the same point.
Ellipse as a parametric curve Polar and parametric curves x=a\cos\theta,\quad y=b\sin\theta a|b|theta a and b must be positive.
Hyperbola using hyperbolic functions Polar and parametric curves x=a\cosh\theta,\quad y=b\sinh\theta a|b|theta This version covers the right-hand branch. Use x=-a cosh(theta) for the left-hand branch.
Parabola as a parametric curve Polar and parametric curves x=at^2,\quad y=2at a|t a cannot be zero.
Rectangular hyperbola parametric form Polar and parametric curves xy=c^2,\quad x=ct,\quad y=\frac{c}{t} c|t t cannot be zero.
Point traced by rotating rods Polar and parametric curves x=L_1\cos(\omega_1t)+L_2\cos(\omega_2t),\quad y=L_1\sin(\omega_1t)+L_2\sin(\omega_2t) L1|L2|omega1|omega2|t_min|t_max This is a generalised build from the problem style; keep as optional advanced tool.
Tangent to a circle using parameter theta Polar and parametric curves x\cos\theta_1+y\sin\theta_1=a a|theta1 Angle units must be clear.
Tangent to an ellipse using parameter theta Polar and parametric curves \frac{x\cos\theta_1}{a}+\frac{y\sin\theta_1}{b}=1 a|b|theta1 a and b must be positive; angle units must be clear.
Tangent to hyperbola using parameter theta Polar and parametric curves \frac{x}{a}\cosh\theta_1-\frac{y}{b}\sinh\theta_1=1 a|b|theta1 As theta grows large, the tangent approaches an asymptote.
Tangent to a parametric parabola Polar and parametric curves ty=x+at^2 a|t a cannot be zero.
Add and subtract complex numbers Complex numbers (a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i a|b|c|d|operation None beyond numeric parsing.
Complex conjugate Complex numbers \overline{x+iy}=x-iy x|y For real numbers the conjugate is unchanged.
Divide complex numbers using conjugates Complex numbers \frac{z_1}{z_2}=\frac{z_1\overline{z_2}}{z_2\overline{z_2}} a|b|c|d Cannot divide by 0+0i.
Multiply complex numbers in rectangular form Complex numbers (a+bi)(c+di)=(ac-bd)+(bc+ad)i a|b|c|d Remember i²=-1.
Complex locus from argument equation Complex numbers \arg(z-z_0)=\theta z0_real|z0_imaginary|theta Point z0 itself is excluded because argument is undefined there.
Complex locus from modulus equation Complex numbers |z-z_0|=r z0_real|z0_imaginary|r r must be non-negative.
Complex modulus inequality regions Inequalities |z-z_0|\le r z0_real|z0_imaginary|r|inequality_type Boundary inclusion depends on <=, <, >=, >.
Points closer to one complex point than another Complex numbers |z-z_1|<|z-z_2| z1_real|z1_imaginary|z2_real|z2_imaginary If z1=z2, the comparison is meaningless or always equal.
Complex number in rectangular form Complex numbers z=x+iy x|y If y=0 the number is real; if x=0 it is purely imaginary.
Argument of a complex number Polar and parametric curves \arg(z)=\operatorname{atan2}(y,x) x|y|angle_unit Argument is undefined at z=0.
Modulus of a complex number Polar and parametric curves |z|=\sqrt{x^2+y^2} x|y Modulus is always non-negative.
Polar to rectangular form Polar and parametric curves x=r\cos\theta,\quad y=r\sin\theta r|theta|angle_unit Angle units must be clear.
Rectangular to polar form Polar and parametric curves z=x+iy=r(\cos\theta+i\sin\theta) x|y z=0 has modulus 0 and no unique argument.
Divide complex numbers in polar form Polar and parametric curves \frac{r_1\angle\theta_1}{r_2\angle\theta_2}=\frac{r_1}{r_2}\angle(\theta_1-\theta_2) r1|theta1|r2|theta2 r2 cannot be zero.
Multiply complex numbers in polar form Polar and parametric curves r_1\angle\theta_1\times r_2\angle\theta_2=r_1r_2\angle(\theta_1+\theta_2) r1|theta1|r2|theta2 Angle wrapping should be clear.
Complex conjugate roots of real polynomial equations Complex numbers p(z)=0,\; p \text{ real coefficients},\; z=a+bi \Rightarrow \overline z=a-bi \text{ is also a root} root_real|root_imaginary Only guaranteed for polynomial equations with real coefficients.
Angular spacing of complex roots Complex numbers \Delta\theta=\frac{2\pi}{n} n n must be positive.
Roots of a complex number Complex numbers z_k=r^{1/n}\left(\cos\frac{\theta+2k\pi}{n}+i\sin\frac{\theta+2k\pi}{n}\right) r|theta|n n must be a positive integer; roots are evenly spaced around a circle.
General nth roots of a complex number Complex numbers z_k=r^{1/n}\left(\cos\frac{\theta+2k\pi}{n}+i\sin\frac{\theta+2k\pi}{n}\right) r|theta|n|k n must be a positive integer; k runs from 0 to n-1; r must be non-negative.
Roots of unity Complex numbers z_k=\cos\frac{2k\pi}{n}+i\sin\frac{2k\pi}{n} n n must be a positive integer.
Complex exponential form Complex numbers e^{i\theta}=\cos\theta+i\sin\theta theta Angle units must be clear.
De Moivre's theorem for powers Complex numbers (\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta) theta|n For integer n this is straightforward; rational powers need branch awareness.
Capacitive reactance Complex numbers X_C=\frac{1}{\omega C} omega|C omega and C must be positive.
Inductive reactance Complex numbers X_L=\omega L omega|L L and omega should be non-negative.
Represent a sinusoidal quantity as a phasor Complex numbers x(t)=X_0\cos(\omega t+\phi)=\Re\{X_0e^{i(\omega t+\phi)}\} amplitude|omega|phase|time Phase may be entered in degrees or radians.
Series RLC resonance condition Complex numbers \omega_0=\frac{1}{\sqrt{LC}} L|C L and C must be positive.
Current amplitude and phase in a series RLC circuit Series and approximation I(t)=\frac{E_0\cos(\omega t+\phi)}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}} E0|R|L|C|omega|time Requires valid impedance; resonance occurs when omegaL = 1/(omegaC).
Series RLC impedance Series and approximation Z=R+i\left(\omega L-\frac{1}{\omega C}\right) R|L|C|omega omega and C must be positive; R and L should be non-negative.
3D vector component form Vectors and 3D geometry \mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k} Fx|Fy|Fz Zero vector has no unique direction.
Direction cosines of a 3D vector Vectors and 3D geometry \cos\alpha=\frac{l}{r},\quad \cos\beta=\frac{m}{r},\quad \cos\gamma=\frac{n}{r} l|m|n Zero vector has no direction cosines.
Direction cosine identity Vectors and 3D geometry \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 cos_alpha|cos_beta|cos_gamma Inputs must satisfy squares summing to 1 within tolerance.
Magnitude of a 3D vector Vectors and 3D geometry |\mathbf{F}|=\sqrt{F_x^2+F_y^2+F_z^2} Fx|Fy|Fz Magnitude is non-negative.
Resultant of two forces Vectors and 3D geometry R=\sqrt{F_1^2+F_2^2+2F_1F_2\cos\theta} F1|F2|theta_between Forces should be non-negative; angle unit must be clear.
Work done as scalar product Vectors and 3D geometry W=\mathbf{F}\cdot\mathbf{d}=|\mathbf{F}||\mathbf{d}|\cos\theta force_magnitude|displacement_magnitude|angle Units should be shown but not enforced.
Unit vector in the same direction Vectors and 3D geometry \hat v=\frac{\vec v}{|\vec v|} x|y Zero vector has no unit vector.
Represent a 2D vector Vectors and 3D geometry \vec v=(x,y) x|y Zero vector has no unique direction.
Magnitude of a 2D vector Vectors and 3D geometry |\vec v|=\sqrt{x^2+y^2} x|y Magnitude is always non-negative.
Area of parallelogram using cross product Vectors and 3D geometry \text{Area}=|\mathbf{A}\times\mathbf{B}| A_components|B_components Area is zero if vectors are parallel or one vector is zero.
Area of triangle using cross product Vectors and 3D geometry \text{Area}=\frac{1}{2}|\mathbf{AB}\times\mathbf{AC}| point_A|point_B|point_C Area is zero for collinear points.
Cross product by components Vectors and 3D geometry \mathbf{A}\times\mathbf{B}=(A_2B_3-A_3B_2)\mathbf{i}+(A_3B_1-A_1B_3)\mathbf{j}+(A_1B_2-A_2B_1)\mathbf{k} A1|A2|A3|B1|B2|B3 Works only for 3D vectors.
Vector product / cross product magnitude and direction Vectors and 3D geometry |\mathbf{A}\times\mathbf{B}|=|\mathbf{A}||\mathbf{B}|\sin\theta A_components|B_components If vectors are parallel, cross product is zero. Direction is undefined for zero result.
Cross product unit-vector rules Vectors and 3D geometry \mathbf{i}\times\mathbf{j}=\mathbf{k},\quad \mathbf{j}\times\mathbf{k}=\mathbf{i},\quad \mathbf{k}\times\mathbf{i}=\mathbf{j} unit_vector_pair Order matters; reversing order changes the sign.
Moment / torque as a cross product Vectors and 3D geometry \boldsymbol{\tau}=\mathbf{r}\times\mathbf{F} r_components|F_components Torque is zero if force is parallel to displacement vector.
Angle between two vectors Vectors and 3D geometry \cos\theta=\frac{\mathbf{A}\cdot\mathbf{B}}{|\mathbf{A}||\mathbf{B}|} A_components|B_components Angle undefined if either vector is zero.
Scalar product / dot product angle form Vectors and 3D geometry \mathbf{A}\cdot\mathbf{B}=|\mathbf{A}||\mathbf{B}|\cos\theta A_components|B_components Angle undefined if either vector is zero.
Scalar product / dot product by components Vectors and 3D geometry \mathbf{A}\cdot\mathbf{B}=A_xB_x+A_yB_y+A_zB_z Ax|Ay|Az|Bx|By|Bz Works in 2D by setting z components to zero.
Orthogonal vectors checker Vectors and 3D geometry \mathbf{A}\cdot\mathbf{B}=0 A_components|B_components Zero vector needs explanatory warning.
Scalar projection of A onto B Vectors and 3D geometry \operatorname{comp}_{\mathbf{B}}\mathbf{A}=\frac{\mathbf{A}\cdot\mathbf{B}}{|\mathbf{B}|} A_components|B_components Cannot project onto zero vector.
Vector projection of A onto B Vectors and 3D geometry \operatorname{proj}_{\mathbf{B}}\mathbf{A}=\frac{\mathbf{A}\cdot\mathbf{B}}{|\mathbf{B}|^2}\mathbf{B} A_components|B_components Cannot project onto zero vector.
Cosine rule from scalar product Vectors and 3D geometry c^2=a^2+b^2-2ab\cos C a|b|included_angle Lengths should be non-negative; angle must be valid.
Pythagoras as a special case of scalar product Vectors and 3D geometry \mathbf{a}\cdot\mathbf{b}=0\Rightarrow c^2=a^2+b^2 a|b Only applies for perpendicular components.
Parallelogram law of vector addition Vectors and 3D geometry \vec R=\vec A+\vec B A_x|A_y|B_x|B_y None beyond zero vector cases.
Closed polygon vector condition Vectors and 3D geometry \mathbf{F}_1+\mathbf{F}_2+\cdots+\mathbf{F}_n=\mathbf{0} list_of_vectors Allow tolerance for decimal inputs.
Triangle law of vector addition Vectors and 3D geometry \vec A+\vec B=\vec C A_x|A_y|B_x|B_y None beyond zero vector cases.
Angle between two straight lines using normal vectors Vectors and 3D geometry \cos\theta=\frac{\mathbf{n}_1\cdot\mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|} line1_a|line1_b|line1_c|line2_a|line2_b|line2_c Invalid line if both a and b are zero.
Line perpendicular to a given 2D direction vector Vectors and 3D geometry ax+by=c direction_a|direction_b|point_x|point_y Direction vector cannot be zero.
Equation of a plane from point and normal vector Vectors and 3D geometry (\mathbf{r}-\mathbf{b})\cdot\mathbf{a}=0 point_bx|point_by|point_bz|normal_ax|normal_ay|normal_az Normal vector cannot be zero.
Point-to-line distance using a perpendicular unit vector Vectors and 3D geometry d=|\overrightarrow{MP}\cdot\hat{p}| point_A|line_point_M|line_normal_vector Normal vector cannot be zero.
Perpendicular distance from point to plane Vectors and 3D geometry d=\frac{|a x_0+b y_0+c z_0+d|}{\sqrt{a^2+b^2+c^2}} plane_a|plane_b|plane_c|plane_d|point_x|point_y|point_z Plane normal cannot be zero.
Unit vector perpendicular to a plane Vectors and 3D geometry \hat{n}=\frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}} plane_a|plane_b|plane_c Normal vector cannot be zero.
Parallel vectors checker Vectors and 3D geometry \mathbf{A}=\lambda\mathbf{B} A_components|B_components Zero vector handling requires care.
Add two 2D vectors Vectors and 3D geometry (x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2) x1|y1|x2|y2 None beyond zero-vector direction handling.
Multiply a vector by a scalar Vectors and 3D geometry \lambda(x,y)=(\lambda x,\lambda y) x|y|lambda Negative scalar reverses direction; zero scalar gives zero vector.
Subtract two 2D vectors Vectors and 3D geometry (x_1,y_1)-(x_2,y_2)=(x_1-x_2,y_1-y_2) x1|y1|x2|y2 None beyond zero-vector direction handling.
Collinearity using vectors Vectors and 3D geometry \vec{AB}=\lambda\vec{AC} point_A|point_B|point_C Repeated points require clear explanation.
Vector equation of a line through two points Vectors and 3D geometry \mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a}) point_a|point_b|t If both points are the same, no unique line is defined.
Midpoint using vectors Vectors and 3D geometry \mathbf{m}=\frac{\mathbf{a}+\mathbf{b}}{2} point_a|point_b Works in 2D and 3D.
Vector equation of a straight line Vectors and 3D geometry \mathbf{r}=\mathbf{a}+t\mathbf{u} point_a_components|direction_u_components|t Direction vector cannot be zero.
Position vector of a point Vectors and 3D geometry \vec{OP}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k} x|y|z_optional For 2D, omit z or set z=0. The origin has zero position vector.
Point dividing a line segment in a ratio Vectors and 3D geometry \mathbf{p}=\frac{n\mathbf{a}+m\mathbf{b}}{m+n} point_a|point_b|ratio_m_to_n m+n cannot be zero. For internal division, m and n should be positive.
Vector from one point to another Vectors and 3D geometry \vec{AB}=\vec{OB}-\vec{OA} Ax|Ay|Az_optional|Bx|By|Bz_optional If A and B are the same point, the vector is zero.
Coplanar vectors checker Vectors and 3D geometry \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})=0 A_components|B_components|C_components Use numerical tolerance for decimal inputs.
Scalar triple product Vectors and 3D geometry \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) A_components|B_components|C_components Zero result if any vector is zero, if two vectors are parallel, or if all three are coplanar.
Scalar triple product as determinant Vectors and 3D geometry [\mathbf{A},\mathbf{B},\mathbf{C}]=\begin{vmatrix}A_1&A_2&A_3\\B_1&B_2&B_3\\C_1&C_2&C_3\end{vmatrix} A_components|B_components|C_components Near-zero determinant should use tolerance.
Vector triple product Vectors and 3D geometry \mathbf{A}\times(\mathbf{B}\times\mathbf{C})=(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{A}\cdot\mathbf{B})\mathbf{C} A_components|B_components|C_components Order and brackets matter.
Linear equation in two variables Linear systems ax+by=c a|b|c If a=0 and b=0, equation is invalid unless c=0, in which case it is an identity.
General system of n linear equations Linear systems \sum_{j=1}^{n}a_{ij}x_j=b_i coefficient_matrix_A|right_hand_side_b Can be unique, inconsistent, or underdetermined; numerical tolerance is needed for decimal input.
2x2 system solution for x using determinants Linear systems x=\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1} a1|b1|c1|a2|b2|c2 Denominator cannot be zero for a unique solution.
2x2 system solution for y using determinants Linear systems y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1} a1|b1|c1|a2|b2|c2 Denominator cannot be zero for a unique solution.
Determinant of a 2x2 coefficient matrix Linear systems \Delta=a_1b_2-a_2b_1 a1|b1|a2|b2 If determinant is zero, the two equations are dependent or inconsistent.
Pair of simultaneous linear equations Linear systems \begin{cases}a_1x+b_1y=c_1\\a_2x+b_2y=c_2\end{cases} a1|b1|c1|a2|b2|c2 System may have one solution, no solution, or infinitely many solutions.
Coincident lines have infinitely many solutions Linear systems \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} a1|b1|c1|a2|b2|c2 Use robust proportionality checks when coefficients include zeros.
Parallel distinct lines have no solution Linear systems \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2} a1|b1|c1|a2|b2|c2 Use determinant/classification logic rather than direct ratios when coefficients include zeros.
3x3 determinant expansion Linear systems \begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33} a11|a12|a13|a21|a22|a23|a31|a32|a33 Large values can overflow in naive implementations; decimal tolerance needed for near-zero checks.
2x2 determinant condition for unique solution Linear systems \begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}=a_1b_2-a_2b_1\ne0 a1|b1|a2|b2 If determinant is zero, system may be inconsistent or dependent.
Zero determinant indicates singular coefficient matrix Linear systems \det(A)=0 matrix_A A zero determinant does not by itself distinguish no solution from infinitely many solutions.
Linear equations from Kirchhoff circuit laws Linear systems A\mathbf{i}=\mathbf{v} circuit_equations_or_matrix This pack does not include a full circuit parser; use equation/matrix input first.
Gauss-Jordan elimination to reduced row echelon form Linear systems \left[A\mid b\right]\rightarrow\left[I\mid x\right] augmented_matrix Requires pivoting or classification when a pivot is zero.
Identity matrix form after Gauss-Jordan elimination Linear systems \left[I\mid x\right] reduced_augmented_matrix Only valid for a unique solution where the coefficient matrix reduces to identity.
Back substitution after triangular reduction Linear systems x_i=\frac{y_i-\sum_{j=i+1}^{n}u_{ij}x_j}{u_{ii}} upper_triangular_matrix_U|right_hand_side_y Diagonal entries must not be zero for direct back substitution.
Gaussian elimination to triangular form Linear systems \left[A\mid b\right]\rightarrow\left[U\mid y\right] augmented_matrix Zero or very small pivots require row swapping or pivoting.
Elementary row replacement operation Linear systems R_i\leftarrow R_i-kR_j target_row|source_row|factor_k Arithmetic may use fractions or decimals; exact fraction mode is useful for learning.
Elementary row scaling operation Linear systems R_i\leftarrow kR_i row_i|scale_factor_k k must not be zero if preserving equivalent equation systems.
Elementary row swap operation Linear systems R_i\leftrightarrow R_j row_i|row_j Needed when a pivot is zero or poor for numerical stability.
Row-sum check during elimination Linear systems s_i=\sum_j a_{ij}+b_i augmented_matrix This is a checking aid, not a proof of correctness.
Detect free variables and infinitely many solutions Linear systems \operatorname{rank}(A)<n\;\text{and system is consistent} row_reduced_augmented_matrix Only applies when the system is consistent.
Detect inconsistent row in row-reduced system Linear systems 0x_1+0x_2+\cdots+0x_n=c,\quad c\ne0 row_reduced_augmented_matrix Use tolerance for decimal entries.
Classify a linear system using ranks Linear systems \operatorname{rank}(A)=\operatorname{rank}([A\mid b]) coefficient_matrix_A|right_hand_side_b Rank calculation needs tolerance for decimal systems.
Residual vector for an approximate solution Vectors and 3D geometry \mathbf{r}=A\mathbf{x}-\mathbf{b} matrix_A|solution_x|right_hand_side_b Small residual suggests consistency with the equations, but scaling affects interpretation.
Condition number indicator Linear systems \kappa(A)=\|A\|\,\|A^{-1}\| matrix_A Requires matrix inverse or numerical library routine; mark as later if not building now.
Ill-conditioned linear system Linear systems \text{small coefficient changes}\Rightarrow\text{large solution changes} matrix_A|right_hand_side_b|rounded_variant_optional A formal condition number may be added later; v1 can use determinant size and perturbation comparison.
Pivoting by selecting a large coefficient Linear systems \text{choose pivot row with largest }|a_{ik}| current_column|candidate_rows If all candidate pivots are zero, the matrix is singular or rank-deficient at that stage.
Diagonal matrix Matrices and determinants a_{ij}=0\quad\text{for }i\ne j diagonal_entries Diagonal matrices are square in the usual definition.
Order of a matrix Matrices and determinants A\in\mathbb{R}^{m\times n} matrix_A Empty rows or jagged rows should be invalid.
Row vector and column vector Vectors and 3D geometry \begin{bmatrix}a_1&a_2&\cdots&a_n\end{bmatrix},\quad\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix} values Length must be at least one.
Zero matrix Matrices and determinants O=\begin{bmatrix}0&0&\cdots&0\\0&0&\cdots&0\\\vdots&\vdots&\ddots&\vdots\end{bmatrix} rows|columns Rows and columns must be positive integers.
Find inverse by reducing all right-hand sides at once Matrices and determinants [A\mid I]\rightarrow[I\mid A^{-1}] square_matrix_A If the left block cannot become I, A is singular.
Find inverse matrix by Gauss-Jordan elimination Linear systems [A\mid I]\rightarrow[I\mid A^{-1}] square_matrix_A If A cannot be reduced to identity, no inverse exists.
Find inverse one column at a time Matrices and determinants A\mathbf{x}_j=\mathbf{e}_j square_matrix_A|unit_vector_ej A must be square and invertible; if any column solve fails, no ordinary inverse exists.
Definition of inverse matrix Matrices and determinants AB=BA=I\Rightarrow B=A^{-1} matrix_A|matrix_B A must be square. Non-square matrices do not have ordinary two-sided inverses.
Uniqueness of inverse matrix Matrices and determinants A^{-1}\text{ is unique if it exists} matrix_A Only applies when the inverse exists.
Verify a proposed inverse Matrices and determinants AA^{-1}=I\quad\text{and}\quad A^{-1}A=I matrix_A|proposed_inverse Both products must be dimensionally defined; A must be square.
Singular matrix has no inverse Matrices and determinants \det(A)=0\Rightarrow A^{-1}\text{ does not exist} square_matrix_A For decimal entries, near-zero determinants need tolerance.
Solve a simultaneous equation system using inverse matrix Matrices and determinants X=A^{-1}B matrix_A|vector_or_matrix_B A must be square and invertible; if singular, use row-reduction classification instead.
Matrix multiplication is associative Matrices and determinants (AB)C=A(BC) matrix_A|matrix_B|matrix_C The products must be dimensionally defined.
Matrix multiplication dimension rule Matrices and determinants (m\times n)(n\times p)=(m\times p) rows_A|cols_A|rows_B|cols_B Multiplication is only defined if columns of A equal rows of B.
Matrix multiplication entry rule Matrices and determinants c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj} matrix_A|matrix_B A columns must equal B rows.
Matrix multiplication is not generally commutative Matrices and determinants AB\ne BA\quad\text{in general} matrix_A|matrix_B AB and BA may not both be defined; even when both are defined, they may have different sizes or entries.
Matrix product can be zero even when neither matrix is zero Matrices and determinants AB=O\nRightarrow A=O\text{ or }B=O matrix_A|matrix_B Requires compatible dimensions.
Matrix addition Matrices and determinants (A+B)_{ij}=a_{ij}+b_{ij} matrix_A|matrix_B Matrices must have the same shape.
Matrix equality Matrices and determinants A=B\iff a_{ij}=b_{ij}\;\text{for all corresponding entries} matrix_A|matrix_B Matrices must have the same shape before comparing entries.
Matrix subtraction Matrices and determinants (A-B)_{ij}=a_{ij}-b_{ij} matrix_A|matrix_B Matrices must have the same shape.
Scalar multiplication of a matrix Matrices and determinants (kA)_{ij}=ka_{ij} scalar_k|matrix_A Any scalar is allowed; zero scalar gives zero matrix of same shape.
Transpose of a matrix Matrices and determinants (A^T)_{ij}=A_{ji} matrix_A Works for any rectangular matrix.
Symmetric matrix Matrices and determinants A^T=A matrix_A Only square matrices can be symmetric in the usual sense.
Transpose of a matrix product Matrices and determinants (AB)^T=B^TA^T matrix_A|matrix_B AB must be defined.
Multiplication by identity matrix Matrices and determinants AI=IA=A matrix_A Identity matrix size must match the side of multiplication.
Postmultiplication by a permutation matrix changes columns Matrices and determinants AP=\text{column-permuted }A matrix_A|permutation_matrix_P P must be a square permutation matrix with compatible size.
Premultiplication by a permutation matrix changes rows Matrices and determinants PA=\text{row-permuted }A matrix_A|permutation_matrix_P P must be a square permutation matrix with compatible size.
Unit or identity matrix Matrices and determinants I_n=\begin{bmatrix}1&0&\cdots&0\\0&1&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&1\end{bmatrix} size_n n must be a positive integer.
Determinant notation for a square matrix Matrices and determinants |A|=\det(A) square_matrix_A Only square matrices have determinants in this chapter's ordinary sense.
Second-order determinant Matrices and determinants \begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc a|b|c|d Works for any numeric entries.
Cofactor of a matrix entry Matrices and determinants C_{ij}=(-1)^{i+j}M_{ij} matrix_A|row_i|column_j Sign alternates by position.
Cofactor sign pattern Matrices and determinants \begin{bmatrix}+&-&+\\-&+&-\\+&-&+\end{bmatrix} size_n Pattern continues for larger determinants.
Expand determinant along any row or column Matrices and determinants |A|=\sum_{j=1}^{n}a_{ij}C_{ij}=\sum_{i=1}^{n}a_{ij}C_{ij} square_matrix_A|chosen_row_or_column Choose a row or column with zeros to reduce work.
Minor of a matrix entry Matrices and determinants M_{ij}=\det(A\text{ with row }i\text{ and column }j\text{ removed}) matrix_A|row_i|column_j For an n x n matrix, the minor matrix has order (n-1) x (n-1).
Third-order determinant expansion along the top row Matrices and determinants |A|=a_{11}\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}-a_{12}\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}+a_{13}\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix} 3x3_matrix_A Use sign pattern + - + across the first row.
Cramer's rule for 2x2 simultaneous equations Linear systems x=\frac{D_x}{D},\quad y=\frac{D_y}{D} a11|a12|a21|a22|b1|b2 If D=0, division is undefined; system may have no solution or infinitely many solutions.
Cramer's rule for 3x3 simultaneous equations Linear systems x_i=\frac{D_i}{D} 3x3_coefficient_matrix_A|rhs_vector_b If D=0, Cramer's rule cannot divide by D. Use system classification instead.
Small determinant can cause numerical sensitivity Linear systems |D|\approx0\Rightarrow\text{solution may be ill-conditioned} coefficient_matrix_A|rhs_vector_b|tolerance Near-zero determinant requires numeric tolerance; exact symbolic inputs may be safer.
Cramer's rule denominator warning Linear systems D=0\Rightarrow \text{Cramer's rule is undefined} coefficient_determinant_D D=0 may mean no solution or infinitely many solutions; Cramer's rule alone does not classify which.
Common factor in a row or column can be factored out Matrices and determinants \det(\ldots,kR_i,\ldots)=k\det(\ldots,R_i,\ldots) matrix_A|row_or_column|factor_k Only one row or column is being scaled for this rule.
Identical rows or columns give zero determinant Matrices and determinants R_i=R_j\Rightarrow |A|=0 matrix_A Also applies to identical columns.
Determinant is linear in a single row or column Matrices and determinants \det(\ldots,R_i+S_i,\ldots)=\det(\ldots,R_i,\ldots)+\det(\ldots,S_i,\ldots) matrix_with_split_row_or_column Only one row or column is split at a time while the rest remains fixed.
Adding a multiple of one row or column to another does not change determinant Matrices and determinants R_i\leftarrow R_i+kR_j\Rightarrow |A|\text{ unchanged} matrix_A|target_row_or_column|source_row_or_column|factor_k The source row or column is not itself changed.
Interchanging two rows or columns changes determinant sign Matrices and determinants R_i\leftrightarrow R_j\Rightarrow |A|\mapsto-|A| matrix_A|row_or_column_swap Each single swap changes the sign once.
Sarrus rule for 3x3 determinants Matrices and determinants \det(A)=a_1b_2c_3+a_2b_3c_1+a_3b_1c_2-a_3b_2c_1-a_1b_3c_2-a_2b_1c_3 3x3_matrix_A Only valid for 3x3 determinants; do not use for larger matrices.
Simplify determinant by creating zeros before expansion Matrices and determinants R_i\leftarrow R_i+kR_j\quad\text{then expand along sparse row/column} matrix_A|row_operations Track row swaps and row scaling separately because they change determinant value.
Triangular determinant equals product of diagonal entries Matrices and determinants \det(U)=\prod_{i=1}^{n}u_{ii} upper_or_lower_triangular_matrix Matrix must be square and triangular.
Adjugate matrix Matrices and determinants \operatorname{adj}(A)=C^T cofactor_matrix_C C must be the cofactor matrix of A.
Inverse using adjugate and determinant Matrices and determinants A^{-1}=\frac{\operatorname{adj}(A)}{|A|},\quad |A|\ne0 square_matrix_A Only works when A is square and det(A) is non-zero.
Cofactor matrix Matrices and determinants C=[C_{ij}],\quad C_{ij}=(-1)^{i+j}M_{ij} square_matrix_A Requires determinant minors for each entry.
Solve a square linear system using the inverse matrix Matrices and determinants AX=B\Rightarrow X=A^{-1}B square_matrix_A|rhs_vector_or_matrix_B Requires A to be non-singular.
Non-singular matrix Matrices and determinants |A|\ne0\Rightarrow A^{-1}\text{ exists} square_matrix_A Numerical tolerance needed for decimal entries.
Singular matrix Matrices and determinants |A|=0\Rightarrow A^{-1}\text{ does not exist} square_matrix_A A near-zero determinant should be flagged as potentially unstable.
Direct determinant method grows very quickly Linear systems \text{determinant work grows roughly with }n! number_of_equations_n This is a rough complexity teaching aid, not an exact runtime model.
Gaussian elimination is more efficient for large systems Linear systems O(n^3) number_of_equations_n Exact operation counts depend on implementation and pivoting.
Inverse method has extra cost compared with elimination for a single system Linear systems AX=B\text{ via }A^{-1}\text{ usually costs more than direct elimination} number_of_equations_n|number_of_rhs_vectors If many right-hand sides share the same A, an inverse/factorization may become more useful.
Diagonal dominance convergence condition Linear systems |a_{ii}|>\sum_{j\ne i}|a_{ij}| coefficient_matrix_A Condition is sufficient but not necessary; failure does not prove divergence.
Gauss-Seidel iteration for three variables Linear systems x_{n+1}=f(y_n,z_n),\quad y_{n+1}=g(x_{n+1},z_n),\quad z_{n+1}=h(x_{n+1},y_{n+1}) three_equations|chosen_rearrangement|initial_guess|iterations Order of updates matters; an unsuitable rearrangement can diverge.
Gauss-Seidel iteration for a two-variable system Linear systems x_{n+1}=f(y_n),\quad y_{n+1}=g(x_{n+1}) two_equations|initial_guess_x0_y0|iterations Uses the newest value as soon as it is available; may still diverge.
Detect divergence in an iterative linear solver Linear systems |x_n|+|y_n|\text{ increasing rapidly}\Rightarrow\text{divergence warning} iteration_sequence|threshold Temporary growth does not always prove divergence; use as a practical warning.
Rounding affects iterative accuracy Linear systems x_n,y_n\text{ rounded each step}\Rightarrow\text{possible accuracy loss} iteration_scheme|decimal_places Rounding at every step can alter convergence behaviour.
Graphical interpretation of iteration Linear systems (x_n,y_n)\to(x_{n+1},y_{n+1}) two_rearranged_equations|initial_guess|iterations Divergent schemes should show points moving away from the solution.
Program logic for Jacobi or Gauss-Seidel iteration Linear systems \max_i|x_i^{(new)}-x_i^{(old)}|<\mathrm{TOL} matrix_A|vector_b|initial_values|tolerance|max_iterations|method Stop if tolerance is reached or maximum iterations are exceeded.
Convergence condition for a rearranged two-equation Jacobi scheme Linear systems |a_1b_2|>|a_2b_1| a1|b1|a2|b2|chosen_rearrangement Condition depends on which variable each equation is rearranged for.
Jacobi iteration for a two-variable system Linear systems x_{n+1}=f(y_n),\quad y_{n+1}=g(x_n) two_equations|initial_guess_x0_y0|iterations May converge, oscillate, or diverge depending on rearrangement.
Comparison of iterative and direct methods Linear systems \text{direct methods: finite solve};\quad\text{iteration: repeated approximation} system_size|sparsity|convergence_condition|required_accuracy Iterative methods are only useful when convergence is reliable.
Sparse matrix advantage in iterative methods Linear systems \text{sparse }A\Rightarrow\text{fewer non-zero operations} coefficient_matrix_A Sparse alone is not enough; convergence must still be checked.
Cobweb diagram for fixed-point iteration Iterative methods y=F(x),\quad y=x iteration_function_F|initial_value|iterations May show oscillation, divergence, trapping cycles, or convergence to an unintended root.
Fixed-point error scale factor Limits and errors e_{n+1}\approx F'(a)e_n iteration_function_F|estimated_root_a|current_error Only a local approximation near the root; poor if x_n is far from the root.
Local convergence condition for fixed-point iteration Iterative methods |F'(a)|<1 iteration_function_F|root_or_estimated_root_a Sufficient local test, not a global guarantee; convergence can still depend on starting value.
Compare rearrangements by derivative scale factor Iterative methods x=F_1(x),\ x=F_2(x),\ldots\quad\Rightarrow\quad |F_i'(a)| candidate_rearrangements|estimated_roots Derivative near a root may not describe behaviour far from the starting point.
Floating sphere depth equation Root finding \frac{1}{3}(3ah^2-h^3) sphere_radius_a|density_ratio|initial_guess|tolerance Physical bounds must be enforced: 0<h<a or appropriate stated interval.
Freudenstein linkage equation solver Root finding \frac{D}{C}\cos\theta-\frac{D}{A}\cos\phi+\frac{D^2+A^2-B^2+C^2}{2AC}-\cos(\theta-\phi)=0 A|B|C|D|theta|initial_phi|tolerance Multiple configurations may exist; geometry validation needed.
Kepler's equation root solver Root finding M=x-E\sin x M|E|initial_guess|tolerance Units must be clear; eccentricity range should be validated.
Newton method for nth roots Iterative methods x_{n+1}=\frac{(r-1)x_n+\frac{A}{x_n^{r-1}}}{r} A|r|initial_guess|tolerance Invalid combinations for real roots must be handled; x_n must avoid zero.
Newton method for square roots Iterative methods x_{n+1}=\frac{1}{2}\left(x_n+\frac{A}{x_n}\right) A|initial_guess|tolerance A must be positive for real square roots; x_n must not be zero.
Resolving two close roots by first finding the nearby stationary point Differentiation f'(x^*)=0,\quad d\approx\sqrt{\frac{-2f(x^*)}{f''(x^*)}},\quad x\approx x^*\pm d function_f|first_derivative|second_derivative|stationary_guess Requires f''(x*) not zero and the expression under the square root to be positive for real close roots.
Newton-Raphson as a fixed-point iteration Root finding F(x)=x-\frac{f(x)}{f'(x)} function_f|derivative_f_prime Requires f'(x) not equal to zero at evaluated points.
Newton-Raphson failure modes Root finding f'(x_n)\approx0\quad\text{or}\quad x_n\to\text{cycle/divergence} function_f|derivative_f_prime|initial_guess Can loop, jump away from a root, hit a stationary point, or converge to a different root.
Multiple roots slow or disturb Newton convergence Root finding f(a)=0,\quad f'(a)=0 function_f|root_candidate Newton may converge only linearly or fail when the derivative vanishes at the root.
Newton-Raphson update Root finding x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} function_f|derivative_f_prime|initial_guess|tolerance|max_iterations Fails if derivative is zero or near zero; can diverge from poor starting values.
Newton update for quadratic example Root finding f(x)=x^2-5x+4,\quad f'(x)=2x-5 initial_guess Starting near derivative zero can be unstable.
Tangent-line interpretation of Newton-Raphson Root finding \text{next estimate} = \text{x-intercept of tangent at }(x_n,f(x_n)) function_f|x_n Tangent may point away from desired root or cross at a poor location.
Second-order convergence when Newton-Raphson works well Root finding e_{n+1}\propto e_n^2 iteration_errors Only applies under favourable conditions near a simple root.
Basic fixed-point iteration Iterative methods x_{n+1}=F(x_n) iteration_function_F|initial_guess|iterations|tolerance Different rearrangements of the same equation can behave very differently.
Bisection interval shrinkage Limits and errors \text{interval width after }n\text{ steps}=\frac{b-a}{2^n} initial_width|iterations|tolerance Only measures interval width, not other modelling error.
Bisection method Root finding m=\frac{a+b}{2} function_f|a|b|tolerance|max_iterations Needs a continuous function and a valid sign-change bracket.
Continuity warning for bracketing methods Root finding f\text{ continuous on }[a,b] function_f|interval A sign change across a discontinuity does not guarantee a root.
Rule of false position Root finding x=\frac{a f(b)-b f(a)}{f(b)-f(a)} function_f|a|b|tolerance|max_iterations Needs a valid bracket; can converge slowly if one endpoint remains fixed.
Fixed-point convergence warning Root finding |F'(r)|<1\Rightarrow\text{local convergence is likely} iteration_function_F|estimated_root This is a local test near a root; it does not guarantee global convergence.
Compare fixed-point rearrangements Root finding f(x)=0\Rightarrow x=F_1(x),\ x=F_2(x),\ldots function_f|candidate_rearrangements|initial_guesses A rearrangement may converge to one root but not another, or may fail from some starting values.
Rewrite root-finding as intersection of two curves Root finding f(x)=0\Leftrightarrow g(x)=h(x) function_f|chosen_split_g_h Different splits can make roots easier or harder to see.
Non-linear equation root-finding Root finding f(x)=0 function_f Closed-form solutions may not exist; numerical methods may be required.
Choose a root-finding method Root finding \text{bracketed root}\to\text{bisection or false position};\quad x=F(x)\to\text{fixed-point test} function_f|known_bracket|initial_guess|required_accuracy Discontinuous functions and multiple roots need extra caution.
Estimate roots by sketching or plotting Root finding y=f(x)\quad\text{root where }y=0 function_f|x_range A rough graph can miss roots or show misleading intersections.
Secant-style replacement rule Root finding x_{n+1}=x_n-y_n\frac{x_n-x_{n-1}}{y_n-y_{n-1}} function_f|x0|x1|iterations|tolerance Can fail if denominator is zero or if approximations move away from the root.
Root bracket from sign change Root finding f(a)f(b)<0\Rightarrow\text{at least one root in }(a,b) function_f|a|b Requires continuity on the interval; discontinuities can create false conclusions.
Tool route map Polynomials \text{factor checks}\rightarrow\text{bracketing}\rightarrow\text{Newton/Birge-Vieta}\rightarrow\text{quadratic factor methods} polynomial_degree|coefficient_type|known_factors|required_accuracy No single method is best for every polynomial.
Factor theorem checker Polynomials p(r)=0\Leftrightarrow (x-r)\text{ is a factor of }p(x) polynomial_coefficients|candidate_root_r Floating-point values need tolerance; exact rational mode is preferable.
Polynomial divided by a linear factor Polynomials p(x)=(x-a)q(x)+r polynomial_p|linear_factor_root_a If r is zero then x-a is an exact factor.
Polynomial equation solving as a special root-finding case Polynomials p(x)=0 polynomial_coefficients Multiple roots and numerical sensitivity must be handled.
Nested polynomial evaluation using Horner form Polynomials p(x)=a_0x^3+a_1x^2+a_2x+a_3=(((a_0x+a_1)x+a_2)x+a_3) polynomial_coefficients|x_value Missing powers must be represented with zero coefficients.
Remainder theorem Polynomials p(x)=(x-a)q(x)+r\quad\Rightarrow\quad p(a)=r polynomial_coefficients|a Use exact arithmetic or tolerance-aware comparison for decimal coefficients.
Synthetic division by a linear factor Polynomials p(x)=(x-r)q(x)+p(r) polynomial_coefficients|linear_factor_root_r If remainder is not zero, x-r is not an exact factor.
Synthetic division recurrence for division by x - x1 Polynomials b_0=a_0,\quad b_r=a_r+b_{r-1}x_1 polynomial_coefficients|x1 The final b value is the remainder; coefficients must be supplied in descending powers.
Worked synthetic division example Polynomials 2x^3+3x^2-10x+6=(x-2)(2x^2+7x+4)+14 polynomial_coefficients=[2,3,-10,6]|divisor_root=2 Remainder 14 means x-2 is not a factor.
Birge-Vieta method for polynomial roots Root finding x_{n+1}=x_n-\frac{p(x_n)}{p'(x_n)} polynomial_coefficients|initial_guess|tolerance|max_iterations May fail near repeated roots or where derivative is very small; use residual checks.
Newton correction using synthetic division values Root finding x_{1}=x_{0}-\frac{p(x_{0})}{p'(x_{0})} polynomial_coefficients|x0 Reject or warn when p'(x0) is zero or very small.
Bairstow's method for quadratic factor extraction Iterative methods x^{2}+cx+d\quad\text{is corrected by solving for }\Delta c,\Delta d polynomial_coefficients|initial_c|initial_d|tolerance|max_iterations Requires solving a small linear correction system; may diverge from poor starting values.
Lin's method for estimating a quadratic factor Iterative methods p(x)=(x^{2}+cx+d)q(x)+(r x+s) polynomial_coefficients|initial_c|initial_d|tolerance|max_iterations Needs a sensible initial quadratic factor estimate; convergence is not guaranteed.
Exact quadratic factor check Polynomials p(x)=(x^{2}+cx+d)q(x)\quad\Leftrightarrow\quad r=0,\ s=0 polynomial_coefficients|quadratic_c|quadratic_d For decimal coefficients use tolerance; display both raw and rounded residuals.
Synthetic division by a quadratic factor Polynomials p(x)=(x^{2}+cx+d)q(x)+(rx+s) polynomial_coefficients|quadratic_c|quadratic_d If the quadratic has leading coefficient not equal to 1, normalise first and scale the quotient as needed.
Complex roots of real-coefficient polynomials occur in conjugate pairs Complex numbers (x-(a+ib))(x-(a-ib))=x^{2}-2ax+(a^{2}+b^{2}) complex_root_a_plus_bi Only guaranteed when all polynomial coefficients are real.
Use derivative behaviour to reason about how many roots exist Differentiation p'(x)=0\quad\text{marks stationary points of }p(x) polynomial_coefficients Stationary points require solving the derivative; exact forms may be unavailable for high degree.
Descartes' rule of signs Root finding N_{+}\leq V(p(x)),\quad N_{-}\leq V(p(-x)) polynomial_coefficients Zero coefficients are ignored; the rule gives upper bounds, not exact counts.
Tabulate values to bracket polynomial roots Root finding p(a)p(b)<0\Rightarrow\text{at least one real root in }(a,b) polynomial_coefficients|x_min|x_max|step Sign-change bracketing can miss even-multiplicity roots that touch the axis.
Practical synthetic division table for division by a linear factor Polynomials p(x)=(x-a)q(x)+r polynomial_coefficients|linear_factor_root_a Use zero coefficients for missing powers; if divisor is kx-c, divide by x-c/k and scale the quotient correctly.
Use a second synthetic division pass to obtain p'(x0) Differentiation p(x)=(x-x_0)q(x)+p(x_0),\quad q(x_0)=p'(x_0) polynomial_coefficients|x0 This works for evaluating the derivative value at x0, not for returning the whole derivative polynomial.
Crank-piston displacement Reference x(\theta)=r\left(1-\cos\theta+m-\sqrt{m^2-\sin^2\theta}\right) r|m|theta r: crank radius; m: l/r connecting rod ratio; theta: crank angle
Crank-piston velocity Reference v=r\omega\left(\sin\theta+\frac{\sin\theta\cos\theta}{\sqrt{m^2-\sin^2\theta}}\right) omega omega: angular speed dtheta/dt
Chain rule Differentiation \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} template values Check domain and approximation conditions.
Circle tangent from implicit differentiation Differentiation xx_1+yy_1=a^2 template values Check domain and approximation conditions.
Parametric derivative Differentiation \frac{dy}{dx}=\frac{dy/dt}{dx/dt} template values Check domain and approximation conditions.
Cycloid slope Polar and parametric curves \frac{dy}{dx}=\frac{\sin\theta}{1-\cos\theta}=\cot\left(\frac{\theta}{2}\right) template values Check domain and approximation conditions.
Stationary point condition Differentiation f'(x)=0 template values Check domain and approximation conditions.
Second-derivative test Differentiation f'(x_0)=0,\ f''(x_0)>0 \Rightarrow ext{local minimum};\quad f''(x_0)<0 \Rightarrow ext{local maximum} template values Check domain and approximation conditions.
Open-box volume from 80 by 40 sheet Reference V(x)=x(80-2x)(40-2x) x x: corner square side length
Parametric slope rule Polar and parametric curves rac{dy}{dx}= rac{dy/dt}{dx/dt} template values Check domain and approximation conditions.
Circle second derivative Differentiation rac{d^2y}{dx^2}=- rac{a^2}{y^3} a a: circle radius in x^2+y^2=a^2
Point of inflection Reference ext{Concavity changes at the point} template values Check domain and approximation conditions.
Leibniz theorem Reference \frac{d^n}{dx^n}(uv)=\sum_{r=0}^{n}{n\choose r}\frac{d^{n-r}u}{dx^{n-r}}\frac{d^r v}{dx^r} template values Check domain and approximation conditions.
Curvature definition Curvature \kappa=\frac{d\psi}{ds} template values Check domain and approximation conditions.
Radius of curvature for y=f(x) Curvature \rho=\frac{\left(1+\left(\frac{dy}{dx}\right)^2\right)^{3/2}}{\frac{d^2y}{dx^2}} template values Check domain and approximation conditions.
Magnitude of radius of curvature Curvature |\rho|=\frac{\left(1+\left(\frac{dy}{dx}\right)^2\right)^{3/2}}{\left|\frac{d^2y}{dx^2}\right|} template values Check domain and approximation conditions.
Centre of curvature Curvature x_c=x-\frac{(1+y'^2)y'}{y''},\qquad y_c=y+\frac{1+y'^2}{y''} template values Check domain and approximation conditions.
Forward difference Reference \Delta f(x_0)=f(x_0+h)-f(x_0) template values Check domain and approximation conditions.
Beam derivative chain Differentiation y\rightarrow y'\rightarrow y''\rightarrow y'''\rightarrow y'''' template values Check domain and approximation conditions.
Potential energy equilibrium condition Reference \frac{dV}{dx}=0 template values Check domain and approximation conditions.
Forward-difference expansion for integer n Reference f(x_0+nh)=f(x_0)+n\Delta f(x_0)+\frac{n(n-1)}{2!}\Delta^2 f(x_0)+\frac{n(n-1)(n-2)}{3!}\Delta^3 f(x_0)+\cdots template values Check domain and approximation conditions.
Newton-Gregory forward interpolation Numerical methods f(x_0+ph)=f_0+p\Delta f_0+\frac{p(p-1)}{2!}\Delta^2f_0+\frac{p(p-1)(p-2)}{3!}\Delta^3f_0+\cdots template values Check domain and approximation conditions.
Shift and difference operator identity Reference 1+\Delta=E template values Check domain and approximation conditions.
Backward difference Reference \nabla f(x_0)=f(x_0)-f(x_0-h) template values Check domain and approximation conditions.
Newton-Gregory backward interpolation Numerical methods f(x_0+ph)=f_0+p\nabla f_0+\frac{p(p+1)}{2!}\nabla^2f_0+\frac{p(p+1)(p+2)}{3!}\nabla^3f_0+\cdots template values Check domain and approximation conditions.
Central difference Reference \delta f\left(x_0+\frac{h}{2}\right)=f(x_0+h)-f(x_0) template values Check domain and approximation conditions.
Lagrange interpolation polynomial Numerical methods P(x)=\sum_{r=1}^{N} f_r\prod_{k=1,\,k\ne r}^{N}\frac{x-x_k}{x_r-x_k} template values Check domain and approximation conditions.
Three-point Lagrange form Numerical methods P(x)=f_1\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}+f_2\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}+f_3\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)} template values Check domain and approximation conditions.
Forward-difference first derivative Differentiation f'(x_0)\approx \frac{1}{h}\left[\Delta f_0-\frac{1}{2}\Delta^2 f_0+\frac{1}{3}\Delta^3 f_0-\frac{1}{4}\Delta^4 f_0+\cdots\right] template values Check domain and approximation conditions.
Backward-difference first derivative Differentiation f'(x_0)\approx \frac{1}{h}\left[\nabla f_0+\frac{1}{2}\nabla^2 f_0+\frac{1}{3}\nabla^3 f_0+\frac{1}{4}\nabla^4 f_0+\cdots\right] template values Check domain and approximation conditions.
Central first-derivative formula Differentiation f'(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h} template values Check domain and approximation conditions.
Central second-derivative formula Differentiation f''(x_0)\approx \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} template values Check domain and approximation conditions.
Rolle's theorem Reference f(a)=f(b),\ f\text{ continuous on }[a,b],\ f\text{ differentiable on }(a,b)\Rightarrow \exists \xi\in(a,b): f'(\xi)=0 template values Check domain and approximation conditions.
Mean Value Theorem Reference \exists \xi\in(a,b): f'(\xi)=\frac{f(b)-f(a)}{b-a} template values Check domain and approximation conditions.
Tangent approximation Reference f(x_0+h)\approx f(x_0)+h f'(x_0) template values Check domain and approximation conditions.
Chord approximation Reference y=f(x_0)+\frac{f(x_0+h)-f(x_0)}{h}(x-x_0) template values Check domain and approximation conditions.
Fixed-point local error relation Limits and errors e_{n+1}\approx F'(a)e_n e_n|a e_n: current error x_n-a; a: fixed point with a=F(a)
Fixed-point local convergence condition Reference |F'(a)|<1 \Rightarrow \text{local convergence} template values Check domain and approximation conditions.
Newton–Raphson iteration Iterative methods x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} template values Check domain and approximation conditions.
Newton–Raphson quadratic error model Limits and errors e_{n+1}\approx -\frac{1}{2}\frac{f''(x_n)}{f'(x_n)}e_n^2 template values Check domain and approximation conditions.
Quadratic approximation Reference f(x)\approx f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2 template values Check domain and approximation conditions.
Geometric series sum Series and approximation a+ar+ar^2+\cdots = \frac{a}{1-r}\quad (|r|<1) template values Check domain and approximation conditions.
Divergence criterion for series Series and approximation u_n\not\to 0 \Rightarrow \sum u_n\ \text{diverges} template values Check domain and approximation conditions.
Ratio test Reference L=\lim_{n\to\infty}\left|\frac{u_{n+1}}{u_n}\right|,\ L<1\Rightarrow \text{absolute convergence},\ L>1\Rightarrow \text{divergence} template values Check domain and approximation conditions.
General power series Series and approximation a_0+a_1x+a_2x^2+a_3x^3+\cdots template values Check domain and approximation conditions.
Radius of convergence idea Reference |x|<R\Rightarrow \text{convergence},\quad |x|>R\Rightarrow \text{divergence} template values Check domain and approximation conditions.
Maclaurin series Series and approximation f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\frac{x^3}{3!}f'''(0)+\cdots template values Check domain and approximation conditions.
Sine Maclaurin series Series and approximation \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots template values Check domain and approximation conditions.
Cosine Maclaurin series Series and approximation \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots template values Check domain and approximation conditions.
Exponential Maclaurin series Series and approximation e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots template values Check domain and approximation conditions.
Hyperbolic cosine series Series and approximation \cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots template values Check domain and approximation conditions.
Hyperbolic sine series Series and approximation \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots template values Check domain and approximation conditions.
Logarithmic series Series and approximation \log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots template values Check domain and approximation conditions.
Binomial expansion Reference (1+x)^s=1+sx+\frac{s(s-1)}{2!}x^2+\frac{s(s-1)(s-2)}{3!}x^3+\cdots template values Check domain and approximation conditions.
Series stopping rule Series and approximation |u_n|<\mathrm{EPS}\ \Rightarrow\ \text{stop summing} template values Check domain and approximation conditions.
Taylor series about x=a Series and approximation f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+\frac{(x-a)^3}{3!}f'''(a)+\cdots a|x a: expansion centre; x: target value
Taylor polynomial of degree n Series and approximation T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k template values Check domain and approximation conditions.
First-order Taylor approximation Series and approximation f(a+h)\approx f(a)+hf'(a) template values Check domain and approximation conditions.
Second-order Taylor approximation Series and approximation f(a+h)\approx f(a)+hf'(a)+\frac{h^2}{2}f''(a) template values Check domain and approximation conditions.
Central difference for first derivative Differentiation f'(a)\approx\frac{f(a+h)-f(a-h)}{2h} template values Check domain and approximation conditions.
Central difference for second derivative Differentiation f''(a)\approx\frac{f(a+h)-2f(a)+f(a-h)}{h^2} template values Check domain and approximation conditions.
Radius of curvature at origin with horizontal tangent Curvature \rho=\frac{1}{f''(0)} template values Check domain and approximation conditions.
L’Hôpital’s rule Reference \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} template values Check domain and approximation conditions.
Small absolute error estimate Limits and errors \delta f\approx f'(x)\,\delta x template values Check domain and approximation conditions.
Small relative error estimate Limits and errors \frac{\delta f}{f(x)}\approx\frac{f'(x)}{f(x)}\delta x template values Check domain and approximation conditions.
Percentage error estimate Limits and errors \%\,\text{error}\approx\left|\frac{f'(x)}{f(x)}\delta x\right|\times100 template values Check domain and approximation conditions.
Relative error for product-power formula Limits and errors Q=Cx^ay^bz^c\Rightarrow \frac{\delta Q}{Q}\approx a\frac{\delta x}{x}+b\frac{\delta y}{y}+c\frac{\delta z}{z} template values Check domain and approximation conditions.
Simple pendulum period Reference T=2\pi\sqrt{\frac{l}{g}} template values Check domain and approximation conditions.
Pendulum percentage error relation Limits and errors \frac{\delta g}{g}\approx\frac{\delta l}{l}-2\frac{\delta T}{T} template values Check domain and approximation conditions.
First non-zero derivative stationary test Differentiation f'(a)=\cdots=f^{(n-1)}(a)=0,\ f^{(n)}(a)\ne0 template values Check domain and approximation conditions.
Even-order stationary point rule Differentiation n\ \text{even}:\ f^{(n)}(a)>0\Rightarrow\min,\quad f^{(n)}(a)<0\Rightarrow\max template values Check domain and approximation conditions.
Odd-order stationary point rule Differentiation n\ \text{odd}\Rightarrow\text{not a local maximum/minimum in the usual smooth case} template values Check domain and approximation conditions.
Simple one-dimensional search step Reference x_{new}=x_{old}+h template values Check domain and approximation conditions.
Search stopping condition Reference |h|<\mathrm{TOL}\quad\text{or}\quad b-a<\mathrm{TOL} template values Check domain and approximation conditions.