# Math 21a Fall 2019

## Multivariable Calculus

# Two vector Fields

## F = [x^{3} - x, y^{3} - y ]

This is a gradient field. You can see that from the formula
or that Q_{x}- P

_{y}= 0. A potential is f(x,y) = x

^{4}/4 - x

^{2}/2 + y

^{4}/4 - x

^{y}/2.

## F = [y^{3} - y, x^{3} - x ]

This is not a gradient field. You see that the curl Q_{x}- P

_{y}= 3x^2-3y^2 is nonzero. You see that if x is bigger than y, then the curl is positive and you see the vector field turning counter clockwise there.. If y is bigger than x, you see that the curl is negative. The vector field turns clockwise.