Engineering Board Exam Reviewer · Vector Calculus: Gradient, Divergence, and Curl · hard

The field F=(2xy+z3), i+x2, j+3xz2, k is conservative. Find a scalar potential with F= .

1. x2 y + z3
2. x2 y + x z2
3. x2 y + x z3
4. x y + x z3

Answer

C. x2 y + x z3

Solution

Integrate / x=2xy+z3: =x2y+xz3+g(y,z). Then / y=x2+gy=x2 gy=0, and / z=3xz2+gz=3xz2 gz=0. So =x2y+xz3 (plus a constant).

More Vector Calculus: Gradient, Divergence, and Curl questions

1. Find the directional derivative of =x2+yz at (1,2,1) in the direction of i+2 j+2 k.
2. Find × F if F=y2z, i+xz2, j+x2y, k.
3. Find · F if F=x2y, i+yz2, j+xz, k.
4. Find if =x2y+z.
5. Evaluate C(x,dy-y,dx) counterclockwise around the circle x2+y2=4.

All Vector Calculus: Gradient, Divergence, and Curl questions