Vector Calculus Question! Need help please.?

Question

F is a vector field, f is a scalar field

One of the definitions given: ( f F) (x,y,z) = f(x,y,z)F(x,y,z)

So I have to prove these:

curl(fF) = fcurlF + (del f) x F

and

curl(curlF) = grad(divF) - del^2 F

Thanks.

Answer 1

First look at the x-component. The x-component of curl(A) is ∂Az/∂y - ∂Ay/∂z. If A is the product of a scalar field times a vector field, only vector field has x, y, and z components. Therefore the x-component of the curl becomes

∂(f*Fz)∂y - ∂(f*Fy)/∂z

Expanding the derivatives (product rule):

f*∂Fz/∂y + ∂f/∂y*Fz - f*∂Fy/∂z - ∂f/∂z*Fy

Rearrange the terms:

f*(∂Fz/∂y - ∂Fy/∂z) + ∂f/∂y*Fz - ∂f/∂z*Fy

The first pair is clearly the x component of f*curlF; you need to show the second is the result of (del f) cross F.

del f = ∂f/∂x*i + ∂f/∂y*j + ∂f/∂z*k

and

F = Fx*i + Fy*j * Fz*k

The x-component general cross product is given by

a × b = (aybz − azby) * i; use ∂f/∂y for ay, ∂f/∂z for az, Fz for bz and Fy for by to get the the x-component as

∂f/∂y * Fz - ∂f/∂z * Fy

Again look at the x-component (coefficient of unit vector i):
this is clearly the second pair of the x-component of the curl of the product. You can do this for each of the components, and you will get the answer.

For the second, you just expand the curl as I did above, and expand each derivative, collect terms and identify them as components of grad(divF) and del^2(F).

Find my answers here:

EDIT: These images have been updated 3/3/07

Page 1: http://img126.imageshack.us/img126/1024/delcrossdelcrossf1nf0.png
Page 2: http://img180.imageshack.us/img180/7798/delcrossdelcrossf2zk9.png

Answer 2

Calculus 2 has some "in intensity" calculus like countless integration innovations etc., as others have remarked. Vector calculus includes the thank you to handle and do calculus with vectors. it somewhat is user-friendly actual. you have some concepts like Gauss's theorem, green's theorem and a few suggestions like divergence, curl etc., that are exceedingly hassle-free. it form of feels extra valuable to me in case you're taking the vector calculus with purposes direction. you do no longer want Calculus 2 for doing vector calculus effectively. undemanding calculus is sufficient. yet once you're frightened which you will get perplexed in case you circulate away maths for one entire semester then you truly can take calc 2. yet as a suggested till now Vector Calculus is easier and you do no longer want calc 2 for vector calculus.