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Gauss' and Stokes' Theorems

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Please check my work for the following problem:

Homework Statement

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

a) [tex] \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds[/tex]
b) [tex]- \int_{S} \nabla \phi \times ds = \int_{C} \phi dl[/tex]

The attempt at a solution

So I start with 'a' and I'll subsitute: A(r) = c [tex]\phi[/tex](r)

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

right?

So I start with 'b' and I'll subsitute: A(r) = c [tex]\phi[/tex](r)


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

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

Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
41,738
898
Are you aware that
[tex]\nabla \times c\phi= c (\nabla \times \phi)[/tex]?
 

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