Vector Surface and Volume Elements

Student handout: Vector Surface and Volume Elements

AIMS Maxwell 2021

Students use known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.
This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

Small Group Activity schedule 30 min. build Tabletop Whiteboard with markers,Coordinate Axes description Student handout (PDF)

What students learn Students use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.

Find the formulas for the differential surface and volume elements for a plane, for a finite cylinder (including the top and bottom), and for a sphere. Make sure to draw an appropriate figure.

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