# Gauss' and Stokes' Theorems

1. Jan 28, 2007

### indigojoker

Please check my work for the following problem:

The problem statement, all variables and given/known data

By subsituting A(r) = c $$\phi$$(r) in Gauss's and Stokes theorems, where c is an arbitrary constant vector, find these two other "fundamental theorems":

a) $$\int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds$$
b) $$- \int_{S} \nabla \phi \times ds = \int_{C} \phi dl$$

The attempt at a solution

So I start with 'a' and I'll subsitute: A(r) = c $$\phi$$(r)

Original equation:
$$\int_{\tau} (\nabla \cdot A) d \tau = \int_{S} A \cdot ds$$
Subsitution:
$$\int_{\tau} (\nabla \cdot c \phi) d \tau = \int_{S} c \phi \cdot ds$$
$$c \int_{\tau} (\nabla \cdot \phi) d \tau = c \int_{S} \phi \cdot ds$$
this leads us back to the equation that we want:
$$\int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds$$

right?

So I start with 'b' and I'll subsitute: A(r) = c $$\phi$$(r)

Original equation:
$$\int_{s} (\nabla \times A) \cdot ds = \int_{C} A \cdot dl$$
Subsitution:
$$\int_{s} (\nabla \times c \phi) \cdot ds = \int_{C} c \phi \cdot dl$$
$$\int_{s} \nabla \cdot ( c \phi \times ds) = c \int_{C} \phi \cdot dl$$

i am stuck here on what to do for part b.

2. Jan 29, 2007

### HallsofIvy

Staff Emeritus
Are you aware that
$$\nabla \times c\phi= c (\nabla \times \phi)$$?