Multivariable Calculus

Revise

Directional Derivative

D_\hat{u}f=\nabla{f}\cdot \hat{u}=\langle \frac{\partial{f}}{\partial{x}},\frac{\partial{f}}{\partial{y}} \rangle \cdot \langle a,b \rangle

• $\nabla f(x,y)$ - Gradient Vector
• $\hat{u}$ - Unit Vector

If given $\theta$, $\hat{u}=⟨cos\theta ,sin\theta ⟩$

Maximizing and Minimizing Gradient

Compute the magnitude of $\nabla f$

$|\nabla f(a,b)|=|⟨a,b⟩|$

Multiply by scalar, $-1$, to get the gradient with greatest decrease.

Steps to Solve

1. Compute $\nabla f$.
2. Evaluate $\nabla f$ at $P$.
3. If given $\theta$, use $\hat{u}=⟨cos\theta ,sin\theta ⟩$. Otherwise, use $\hat{u}$ to compute the dot product, $\nabla f\cdot \hat{u}$.