Key Equations

Key Equations

Vertical trace	$f\left(a,y\right)=z$ for $x=a$ or $f\left(x,b\right)=z$ for $y=b$
Level surface of a function of three variables	$f(x,y,z)=c$

Partial derivative of $f$ with respect to $x$	$\frac{\partial f}{\partial x}=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}$
Partial derivative of $f$ with respect to $y$	$\frac{\partial f}{\partial y}=\underset{k\to 0}{\text{lim}}\frac{f\left(x,y+k\right)-f\left(x,y\right)}{k}$

Tangent plane	$z=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)$
Linear approximation	$L\left(x,y\right)=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)$
Total differential	$dz=f_x\left(x_0,y_0\right)dx+f_y\left(x_0,y_0\right)dy.$
Differentiability (two variables)	$f\left(x,y\right)=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)+E\left(x,y\right),$ where the error term $E$ satisfies $\underset{\left(x,y\right)\to \left(x_0,y_0\right)}{\text{lim}}\frac{E\left(x,y\right)}{\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2}}=0.$
Differentiability (three variables)	$\begin{array}{cc}f\left(x,y\right)\hfill & =f\left(x_0,y_0,z_0\right)+f_x\left(x_0,y_0,z_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0,z_0\right)\left(y-y_0\right)\hfill \\ & +f_z\left(x_0,y_0,z_0\right)\left(z-z_0\right)+E\left(x,y,z\right),\hfill \end{array}$ where the error term $E$ satisfies $\underset{\left(x,y,z\right)\to \left(x_0,y_0,z_0\right)}{\text{lim}}\frac{E\left(x,y,z\right)}{\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2}}=0.$

Chain rule, one independent variable	$\frac{dz}{dt}=\frac{\partial z}{\partial x}·\frac{dx}{dt}+\frac{\partial z}{\partial y}·\frac{dy}{dt}$
Chain rule, two independent variables	$\frac{dz}{du}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial x}{\partial u}$ $\frac{dz}{dv}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}$
Generalized chain rule	$\frac{\partial w}{\partial t_j}=\frac{\partial w}{\partial x_1}\frac{\partial x_1}{\partial t_j}+\frac{\partial w}{\partial x_2}\frac{\partial x_1}{\partial t_j}+\text{⋯}+\frac{\partial w}{\partial x_m}\frac{\partial x_m}{\partial t_j}$

directional derivative (two dimensions)	$D_{u}f\left(a,b\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\text{cos}\theta ,b+h\text{sin}\theta \right)-f\left(a,b\right)}{h}$ or $D_{u}f\left(x,y\right)=f_x\left(x,y\right)\text{cos}\theta +f_y\left(x,y\right)\text{sin}\theta$
gradient (two dimensions)	$\nabla f\left(x,y\right)=f_x\left(x,y\right)i+f_y\left(x,y\right)j$
gradient (three dimensions)	$\nabla f\left(x,y,z\right)=f_x\left(x,y,z\right)i+f_y\left(x,y,z\right)j+f_z\left(x,y,z\right)k$
directional derivative (three dimensions)	$\begin{array}{cc}\hfill D_{u}f\left(x,y,z\right)& =\nabla f\left(x,y,z\right)·u\hfill \\ & =f_x\left(x,y,z\right)\text{cos}\alpha +f_y\left(x,y,z\right)\text{cos}\beta +f_x\left(x,y,z\right)\text{cos}\gamma \hfill \end{array}$

Discriminant

$D=f_{xx}\left(x_0,y_0\right)f_{yy}\left(x_0,y_0\right)-{\left(f_{xy}\left(x_0,y_0\right)\right)}^2$

Method of Lagrange multipliers, one constraint	$\begin{array}{ccc}\hfill \nabla f\left(x_0,y_0\right)& =\hfill & \lambda \nabla g\left(x_0,y_0\right)\hfill \\ \hfill g\left(x_0,y_0\right)& =\hfill & 0\hfill \end{array}$
Method of Lagrange multipliers, two constraints	$\begin{array}{ccc}\hfill \nabla f\left(x_0,y_0,z_0\right)& =\hfill & {\lambda }_1\nabla g\left(x_0,y_0,z_0\right)+{\lambda }_2\nabla h\left(x_0,y_0,z_0\right)\hfill \\ \hfill g\left(x_0,y_0,z_0\right)& =\hfill & 0\hfill \\ \hfill h\left(x_0,y_0,z_0\right)& =\hfill & 0\hfill \end{array}$