Fundamental Theorems for Vector Fields

In summary, by substituting A(r) = c \phi(r) in Gauss's and Stokes theorems, we can find the following two fundamental theorems: a) \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds b) - \int_{S} \nabla \phi \times ds = \int_{C} \phi dl. This substitution simplifies the equations and leads us back to the desired equations.
  • #1
indigojoker
246
0
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.
 
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  • #2
Are you aware that
[tex]\nabla \times c\phi= c (\nabla \times \phi)[/tex]?
 

What is Gauss' Theorem?

Gauss' Theorem, also known as the Divergence Theorem, states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of that vector field throughout the enclosed volume.

What is Stokes' Theorem?

Stokes' Theorem states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of that vector field over the surface bounded by the curve.

How are Gauss' and Stokes' Theorems related?

Gauss' and Stokes' Theorems are both fundamental theorems in vector calculus that relate a line integral and a surface integral for a vector field. They both relate a closed surface or curve to the properties of the vector field within that surface or curve.

What are the practical applications of Gauss' and Stokes' Theorems?

Gauss' and Stokes' Theorems are used in a wide range of fields including physics, engineering, and mathematics. They are used to calculate flux and circulation of vector fields, and are essential in the study of electromagnetism, fluid dynamics, and other physical phenomena.

Are there any limitations to Gauss' and Stokes' Theorems?

While Gauss' and Stokes' Theorems are powerful tools in vector calculus, they have certain limitations and assumptions. For example, they only apply to smooth vector fields and cannot be used for discontinuous or singular vector fields. Additionally, they only apply to closed surfaces and curves, so they cannot be used for open or infinite surfaces or curves.

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