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Hello,

I'm working through some problems in the Griffith text on electrodynamics. In one of them, the reader is asked to prove the following identity (which is given in the text), which is a generalization (of sorts) on the divergence theorem:

[tex]\large{ \int_V \left(\nabla T\right) dV = \oint_{\partial V}TdA} [/tex]

where T is a scalar field.

I'm not going to go through the proof here (which is relatively straightforward).

Rather, my question simply is: what meaning (physical or otherwise) can be ascribed to the left-hand side of the equation? I can understand taking the volume integral of a scalar field, but grad-T is a vector. How can you take the volume integral of a vector field?

For instance, in the "traditional" divergence theorem, it is the scalar field "div-F" that is integrated through the volume:

[tex]\large{ \int_V \left(\nabla \cdot F\right) dV = \oint_{\partial V} F \cdot dA }[/tex]

But how can a vector field be integrated through a volume?

I'm working through some problems in the Griffith text on electrodynamics. In one of them, the reader is asked to prove the following identity (which is given in the text), which is a generalization (of sorts) on the divergence theorem:

[tex]\large{ \int_V \left(\nabla T\right) dV = \oint_{\partial V}TdA} [/tex]

where T is a scalar field.

I'm not going to go through the proof here (which is relatively straightforward).

Rather, my question simply is: what meaning (physical or otherwise) can be ascribed to the left-hand side of the equation? I can understand taking the volume integral of a scalar field, but grad-T is a vector. How can you take the volume integral of a vector field?

For instance, in the "traditional" divergence theorem, it is the scalar field "div-F" that is integrated through the volume:

[tex]\large{ \int_V \left(\nabla \cdot F\right) dV = \oint_{\partial V} F \cdot dA }[/tex]

But how can a vector field be integrated through a volume?

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