16.1 Videos Guide

16.1a

Definition: (vector field)

• A vector field is a function whose set of inputs are points and whose outputs are vectors.

Exercise:

• Sketch the vector field $\mathbf{F}$.
  $$\mathbf{F}(x,\ y)=\frac{1}{2}x\ \mathbf{i}+y\ \mathbf{j}$$

16.1b

• The gradient as a vector field

Definitions: (conservative vector field and potential function)

• If $\mathbf{F}=\mathbf{\nabla}f$ for some function $f$, then $\mathbf{F}$ is a conservative vector field with potential function $f$

Exercises:

• Find the gradient vector field of $f$.
  $$f(x,\ y,\ z)=x^{2}ye^{\frac{y}{z}}$$

• Find the gradient vector field $\mathbf{\nabla}f$ of $f$ and sketch it.
  $$f(x,\ y)=\frac{1}{2}\left(x^{2}-y^{2}\right)$$