Key Equations

• Directional derivative (two dimensions)
  $D_{\bf{u}} f(a,b)=\underset{h \to {0}}{\lim} \frac{f(a+h\cos{\theta}, b+h\sin{\theta})-f(a,b)}{h}$  or  $D_{\bf{u}} f(x,y)=f_{x}(x,y)\cos{\theta}+f_{y}(x,y)\sin{\theta}$
• Gradient (two dimensions)
  $\nabla f(x,y)=f_{x}(x,y){\bf{i}}+f_{y}(x,y){\bf{j}}$
• Gradient (three dimensions)
  $\nabla f(x,y,z)=f_{x}(x,y,z){\bf{i}}+f_{y}(x,y,z){\bf{j}}+f_{z}(x,y,z){\bf{k}}$
• Directional derivative (three dimensions)
  $\begin{array}{c} D_{\bf{u}} f(x,y,z) &=\nabla f(x,y,z)\cdot{\bf{u}} \hfill \\ \hfill &=f_{x}(x,y,z)\cos{\alpha}+f_{y}(x,y,z)\cos{\beta}+f_{z}(x,y,z)\cos{\gamma}\end{array}$

Glossary

directional derivative

the derivative of a function in the direction of a given unit vector

gradient

the gradient of the function $f(x,y)$ is defined to be $\nabla f(x,y)=(\partial{f}{/}\partial{x}){\bf{i}}+(\partial{f}{/}\partial{y}){\bf{j}}$ which can be generalized to a function of any number of independent variables