Category:Surface Integrals
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This category contains results about Surface Integrals.
Definitions specific to this category can be found in Definitions/Surface Integrals.
Let $S$ be a surface in a vector field $\mathbf F$.
Let $\d S$ be a small element of $S$.
Let $\mathbf v$ be the vector induced by $\mathbf F$ at the middle of $\d S$.
Let $\mathbf {\hat n}$ denote the positive unit normal to $S$ at $\d S$.
Let $\mathbf v$ make an angle $\theta$ with $\mathbf {\hat n}$.
Hence:
- $\mathbf v \cdot \mathbf {\hat n} = v \cos \theta \rd S$
where:
- $\cdot$ denotes dot product
- $v$ denotes the magnitude of $\mathbf v$.
The surface integral of $\mathbf v$ over $S$ is therefore defined as:
- $\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$
Subcategories
This category has only the following subcategory.