- #1
indigojoker
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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.
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.