- #1

Contingency

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## Homework Statement

I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived.

I read that the divergence of a vector field is defined by:

[itex]\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } } [/itex]

Also, that components of the rotor are defined by:

[itex]\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } } [/itex]

I'm trying to understand where the standard "sum of partial derivatives", and the mnemonic determinant with a row of unit vectors come from. I don't see a correlation between the definitions and these simple representations. How are they derived from the definition?

## Homework Equations

[itex]\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } } [/itex]

[itex]\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } } [/itex]

## The Attempt at a Solution

No clue