\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S10

Surface Integrals
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At the end of the course, each student should be able to…

  • S10: SurfInt. Compute and apply surface integrals.

S10: Surface Integrals

  • The net volume of a solid with base given by the surface \(S\) and heights given by \(f(x,y,z)\) at each point is given by the surface integral \(\iint_S f\,d\sigma\).
    • When the surface is parametrized by \(\vect r(u,v)\) with domain \(G\) in the \(uv\) plane, this integral may be computed as \(\iint_G f(\vect r(u,v))\|\vect r_u\times\vect r_v\|\,dA\).
    • The orientation of \(S\) is irrelevant.
  • The flux of the vector field \(\vect F\) passing through the surface \(S\) oriented by \(\vect n\) is given by \(\iint_S \vect F\cdot\vect n\,d\sigma\).
    • When \(\vect r(u,v)\) parametrizes \(S\) with the correct orientation, then this integral may be computed as \(\iint_G \vect F(\vect r(u,v))\cdot(\vect r_u\times\vect r_v)\,dA\).
    • When \(\vect r(u,v)\) parametrizes \(-S\) (with opposite orientation), then this integral may be computed as \(-\iint_G \vect F(\vect r(u,v))\cdot(\vect r_u\times\vect r_v)\,dA\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 15.6 (exercises 1-36)