P(x,y,z)=(ye^xy) i + (xe^xy + e^z) j + (ye^z) k . From choices below which could be a potential function for P

Question

possible choices....

a. f(x, y, z) = e^xy + ye^z
b. f(x, y, z) = e^yz + xe^y
c. f(x, y, z) = e^xz + ze^x
d. f(x, y, z) = e^xy + xe^z
e. f(x, y, z) = e^yz + ze^y
f. f(x, y, z) = e^xz + ye^x
g. f(x, y, z) = e^xy + ze^z
h. f(x, y, z) = e^yz + ye^y
i. f(x, y, z) = e^xz + xe^x
j. none of these

Answer 1

a, F = e^xy +ye^z

dF/dx = ye^xy

dF/dy = xe^xy+e^z

dF/dz = ye^z

Answer 2

a) as
df/dx= y*e^xy
df/dy) x*e^xy+e^z and
df/dz= y*e^z so grad f = P(x,y,z)