Question

Prove each identity, assuming that S and E satisfy the conditions of Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives.
(Note: I haven't gone through all of them yet, but I have found the solutions to the proofs as I'm trying to figure out some myself still, but if you would like to take a attempt first go ahead, and I will post the solution afterwards).

Answer 1

Curl (aka rotor) of any 3D vector field is divergenceless:
div (curl F(r)) =
by components:
div [∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y] =

∂/∂x(∂Fz/∂y - ∂Fy/∂z) +
∂/∂y(∂Fx/∂z - ∂Fz/∂x) +
∂/∂z(∂Fy/∂x - ∂Fx/∂y) =

∂²Fz/∂x∂y - ∂²Fy/∂x∂z +
∂²Fx/∂y∂z - ∂²Fz/∂y∂x +
∂²Fy/∂z∂x - ∂²Fx/∂z∂y =
= zero

According to Gauss theorem surface flux
double integral of curl F dS =
is equal to volume integral of divergence
= triple integral div curl F dV =
= triple integral of ZERO dV =
= zero
Source(s): http://en.wikipedia.org/wiki/Curl
http://en.wikipedia.org/wiki/Divergence_...

Answer 2

\[1. \int\limits \int\limits \textbf{a} \cdotp \textbf{n} dS = 0\] where a is a constant vector.

2. \[V(E) = \frac{ 1 }{ 3 } \int\limits \int\limits_S \textbf{F} \cdotp d\textbf{S}\] where F(x,y,z) = xi++yi+zk (i,j,k are unit vectors)

\[3. \int\limits \int\limits _S curl \textbf{F} \cdotp d\textbf{S} = 0 \]
\[4. \int\limits \int\limits_S D_n f dS = \int\limits \int\limits \int\limits _e \nabla ^2 f dV\]
\[5. \int\limits \int\limits_S (f \nabla g) \cdotp \textbf{n} dS = \int\limits \int\limits \int\limits_E (f \nabla ^2 g+ \nabla f \cdotp \nabla g) dV\]

\[6. \int\limits \int\limits_S (f \nabla g- g \nabla f) \cdot \textbf{n} dS= \int\limits \int\limits \int\limits ( f \nabla ^2 g - g \nabla ^2 f) dV\]

This should be fun, one of them should look familiar to some of you as I've asked it before. Also note solving one may help you with another!

Answer 3

@rational @dan815 @kainui @michele_laino @zarkon tag anyone else you think would be interested as well :)

Answer 4

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