Thread: Vector Calculus

1) Show that the vector field F = (3x^2)i + (5z^2)j + (10yz)k is irrotional. Then find a potential function f(x,y,z) such that gradient f = F.

(Ok, so i got the part of showing that F is irrotional. But when i tried figuring out to find the function i got something weird. I am not sure if i got this right. My answer is: f(x,y,z) = x^3 + 5z^2 + 5yz^2 - 5z^2.)

2) Let F(x,y,z) = (8xz)i + (1-6yz^3)j + (4x^2 - 9y^2z^2)k. Show that integral with lower terminal C of (F.dr) is independent of path by finding a potential function f for F.

If you could please help me with these problems much would be appreciated. Thank you

Originally Posted by davidson89

1) Show that the vector field F = (3x^2)i + (5z^2)j + (10yz)k is irrotional. Then find a potential function f(x,y,z) such that gradient f = F.

(Ok, so i got the part of showing that F is irrotional. But when i tried figuring out to find the function i got something weird. I am not sure if i got this right. My answer is: f(x,y,z) = x^3 + 5z^2 + 5yz^2 - 5z^2.)

[snip]

Your answer is easily checked by taking grad(f) .... and seen to be wrong.

Solve the following system of simultaneous pde's (all derivatives are partial derivatives):

df/dx = 3x^2 .... (1)

df/dy = 5z^2 .... (2)

df/dz = 10yz .... (3)


From (1): f = x^3 + g(y, z).

From (2): dg/dy = 5z^2 => g = 5z^2 y + h(z).

Therefore f = x^3 + 5z^2 y + h(z).

From (3): 10 zy + dh/dz = 10yz => dh/dz = 0 => h = constant. Take h = 0.

Therefore f = x^3 + 5z^2 y.

Originally Posted by davidson89

[snip]
2) Let F(x,y,z) = (8xz)i + (1-6yz^3)j + (4x^2 - 9y^2z^2)k. Show that integral with lower terminal C of (F.dr) is independent of path by finding a potential function f for F.

If you could please help me with these problems much would be appreciated. Thank you

Confirm that curl F = 0, that is rot F = 0. This is sufficient to prove path independence. Nevertheless:

curl F = 0 => F = grad f.

Find f by solving (all derivatives are partial derivatives):

df/dx = 8xz .... (1)

df/dy = 1 - 6yz^3 .... (2)

df/dz = 4x^2 - 9y^2 z^2 .... (3)


From (1): f = 4x^2 z + g(y, z)

From (2): dg/dy = 1 - 6yz^3 => g = y - 3y^2 z^3 + h(z).

Therefore f = 4x^2 z + y - 3y^2 z^3 + h(z).

From (3): 4x^2 - 9 y^2 z^2 + dh/dz = 4x^2 - 9y^2z^2 => dh/dz = 0. Take h = 0.

Therefore f = 4x^2 z + y - 3y^2 z^3.