Vector Calculus II: Summer-2021 03 : Due 21: M 8/9
Around or through; Stokes in action
S0 4268S
Let \(\boldsymbol{\vec{F}}=-y^2\,\boldsymbol{\hat{x}}\).
Compute \(\nabla \times \boldsymbol{\vec{F}}\).
Is \(\boldsymbol{\vec{F}}\) conservative?
Let \(S\) be the portion of the surface defined by \(z=2-y^2\) inside the cylinder \(x^2+y^2=3\). Construct \(d\boldsymbol{\vec{S}}\) for this surface.
Deduce the value of \(\int\!\!\int_S\nabla\times\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{S}}\).
Let \(C\) be the boundary of \(S\). Construct \(d\boldsymbol{\vec{r}}\) for \(C\).
Are your answers to the integrals the same? Should they be?
$d\boldsymbol{\vec{S}}$ and surfaces
S0 4268S
Let \(\boldsymbol{\vec{F}}=x\,\boldsymbol{\hat{y}}\) and \(\boldsymbol{\vec{G}}=-2x\,\boldsymbol{\hat{x}}-2y\,\boldsymbol{\hat{y}}-2z\boldsymbol{\hat{z}}\).
Compute the flux of each vector field through the following surfaces:
\(z=5-x^2\) where \(-2\leq x \leq 2\) and \(-1 \leq y \leq 2\).
The upper hemisphere of radius 4 centimeters.
Flux Fooling
S0 4268S
Let \(D\) be the cylinder with diameter of 10 inches, height of 13 inches including the bottom but not the top. Presume that the \(z\)-axis is through the center of \(D\) parallel to the height.
If \(\boldsymbol{\vec{F}}=-3z\boldsymbol{\hat{z}}\), use geometry to deduce whether the value of \(\int\!\!\int_D\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{S}}\) is positive, negative or zero.
Support your deduction with explicit calculations.
How would your deduction change if the top were included?
Use the Divergence Theorem to write down an equation relating the flux through \(D\) and the flux through the top.
Create a visual equation expressing the same relationship you wrote down for the previous part. That means: draw pictures. Include some color.
Repeat the above steps using the vector field \(\boldsymbol{\vec{G}}=-4x\,\boldsymbol{\hat{x}}+-4y\,\boldsymbol{\hat{y}}\).