20190709

• I-Learn, Problem Set, Class Pictures

Rapid Recall

• Draw the solid whose volume is given by the integral $\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3}\rho^2\sin\phi \,d\rho \,d\phi \,d\theta$.

• Set up an integral formula to compute the $z$-coordinate of the center of mass of the solid above, provided the density is given by $\delta = x^2+y^2+z^2$.

• The temperature at points in the solid above is given by $T(x,y,z) = x+2y$. Set up an integral that would find the average temperature in the solid.

Group problems

1. The spherical change-of-coordinates is given by $$(x,y,z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi).$$
   • Give an equation of the sphere $x^2+y^2+z^2=9$ in spherical coordinates.
   • Give an equation of the cone $x^2+y^2=z^2$ in spherical coordinates.
   • Set up an integral to find the volume of the region in space above the $xy$-plane that is bounded above by the sphere $x^2+y^2+z^2=9$ and below by the cone $z^2=x^2+y^2$. The Jacobian for spherical coordinates is $|\rho^2\sin\phi|$.
   • Give an equation of the plane $z=8$ in spherical coordinates.
   • Set up an integral to find the volume of the region in space above the $xy$-plane that is bounded above by the plane $z=8$ and below by the cone $z^2=x^2+y^2$.
2. Set up an integral formula to compute each of the following:
   1. The average temperature of a metal plate in the $xy$-plane bounded by the curves $y=8-x^2$ and $y=x+2$, where the temperature at points on the plate is given by $f(x,y)=x+y^2$.
   2. The average charge density on a wire that lies along the helix $\vec r(t) = (3\cos t,3\sin t, 4t)$ for $0\leq t\leq 4\pi$, provided the charge at each point on the wire is given by $\sigma(x,y,z) = x^2+y^2+z$.
   3. The average pressure in a solid region in space inside the sphere $x^2+y^2+z^2=9$, provided the pressure at each point in the sphere is given by $p(x,y,z) = 10+x$.

Rapid Recall

• For the vector field $\vec F(x,y,z) = (3xy,2x+y+5z,yz^2)$, compute $\vec \nabla \cdot \vec F$.

• For the vector field $\vec F(x,y,z) = (3xy,2x+y+5z,yz^2)$, compute $\vec \nabla \times \vec F$.

• For the vector field $\vec F(x,y,z) = (3xy,2x+y+5z,yz^2)$ and the curve $\vec r(t) = (t,t^2,t^3)$, set up the work integral $\int_C\vec F\cdot d\vec r$.

Operators

1. What is an operator?
   • The derivative $\frac{df}{dx}$
   • The integral $\int_a^b f(x) dx$
   • The del operator $\vec \nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$ or $\vec \nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$.
2. Three new quantities (A great book: "Div, Grad, Curl, and all that" by H. M. Schey - Electrostatics)
   • Gradient of a function $f$: $\vec \nabla f$.
   • Divergence of a vector field $\vec F$: $\vec \nabla \cdot \vec F$.
   • Curl of a vector field $\vec F$: $\vec \nabla\times \vec F$.

Practice

1. Let $f(x,y,z) = 3xe^{yz^2}$. Compute the gradient $\vec \nabla f$.
2. Let $\vec F(x,y,z) = (3x+4y+5z, xy, yz^2)$. Compute the divergence $\vec \nabla \cdot \vec F$.
3. Let $\vec F(x,y,z) = (3x+4y+5z, xy, yz^2)$. Compute the curl $\vec \nabla \times \vec F$.
4. As a class, let's pick a function $f$ and vector field $\vec F$. Then compute (1) the gradient $\vec \nabla f$, (2) the divergence $\vec \nabla \cdot \vec F$ and (3) the curl $\vec \nabla \times \vec F$.
5. Use the function $f$ from #4, and compute $\vec \nabla f$ followed by $\vec \nabla \times \vec \nabla f$. Simplify your answer.
6. Let $f$ now represent any function. Symbolically compute $\vec \nabla f$ followed by $\vec \nabla \times \vec \nabla f$. Simplify your answer.
7. Use the vector field $\vec F$ from #4. Then compute $\vec \nabla \times \vec F$ followed by $\vec \nabla \cdot ( \vec \nabla \times \vec F)$.
8. Let $\vec F=(M,N,P)$ represent any vector field. Symbolically compute $\vec \nabla \times \vec F$ followed by $\vec \nabla \cdot ( \vec \nabla \times \vec F)$. Simplify your answer.
9. Pick a specific vector field $\vec F$. Then compute $\vec \nabla \cdot \vec F$ followed by $\vec \nabla ( \vec \nabla \cdot \vec F)$.
10. Let $\vec F=(M,N,P)$ represent any vector field. Symbolically compute $\vec \nabla \cdot \vec F$ followed by $\vec \nabla ( \vec \nabla \cdot \vec F)$.

Connecting Gradients and Vector Fields - Potential Functions

1. For $f(x,y)$, recall the gradient is the vector field $\vec \nabla f = (f_x,f_y)$ and the differential is $df = f_xdx+f_ydy = (f_x,f_y)\cdot(dx,dy)$.
2. The derivative of the gradient is $D(\vec \nabla f) = \begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}$.
3. For a vector field $\vec F = (M,N)$, recall the derivative is $D\vec F = \begin{bmatrix}M_{x}&M_{y}\\N_{x}&N_{y}\end{bmatrix}$.
4. The differential of $f$ is $df = f_x dx+f_y dy$. The differential of work done by the vector field $\vec F = (M,N)$ is $dW = Mdx+Ndy$.
   • Physically, what does the integral $\int_C df$ compute? (Add up ... to get ...)
   • Physically, what does the integral $\int_C dW$ compute? (Add up ... to get ...)
   • Suppose $f_x dx+f_y dy = Mdx+Ndy$. then how are $f$ and $\vec F$ related?
   • When $f_x dx+f_y dy = Mdx+Ndy$, then we have $\int_C dW = \int_C df$. What does this mean in words?
   The work done by $\vec F$ along a curve $C$ is equal to ....

Practice

1. Let $\vec F=(2x+3y, 3x+4y)$. Let $C$ be the curve parametrized by $\vec r(t) = (2-2t,3t)$ for $0\leq t\leq 1$.
   • Draw the curve. At a few points on the curve, draw the vector field.
   • From your picture, will the work done by $\vec F$ along $C$ be positive or negative?
   • Compute the work done by $\vec F$ along the curve.
2. Let $f(x,y) = x^2+3xy+2y^2$.
   • Compute $\vec \nabla f$ and $df$.
   • State $\vec r(0)$ and the value of $f$ at $t=0$? Then repeat this at $t=1$.
   • Compute $\int_C df$. In other words, state the total change in $f$ along the curve $C$. Compare this to the first question.
3. Let $f = xy^2+3x$.
   • Compute $\vec \nabla f$.
   • Compute $D^2f$.
   • Compute $\int_{ (2,1) }^{ (-1,3) } df$
4. Let $\vec F = (2xy+4, x^2+2y)$.
   • Compute $D\vec F$.
   • Find a function $f$ so that $\vec F = \vec \nabla f$.
   • Find the work done by $f$ to get from $(2,0)$ to $(0,3)$. (Hint, what is $\int_C df$?)
5. Let $\vec F = (2x+3y, 4x+5y)$.
   • Compute $D\vec F$.
   • Explain why is it impossible to find a function $f$ so that $\vec F = \vec \nabla f$.
6. Under what conditions will a vector field $\vec F$ have a function $f$ such that $\vec\nabla f = \vec F$. We call such function $f$ a potential for $\vec F$.
7. Find a potential for each of the following, or explain why none exists.
   • $\vec F = (2x,3y)$
   • $\vec F = (2y,3x)$
   • $\vec F = (3y,3x)$
   • $\vec F = (4x,5y,6z)$
   • $\vec F = (4x,5z,6y)$
   • $\vec F = (4x,5z,5y)$
   • $\vec F = (2x-y,-x+4y)$
   • $\vec F = (y^2+2x,2xy)$
   • $\vec F = (x+yz,xz+4yz,xy+2y^2)$
   • $\vec F = (x+yz,4yz,xy+2y^2)$
   • $\vec F = (x+yz,xz+4yz,xy)$
   • $\vec F = (yz,xz+4yz,xy+2y^2)$

Boundaries

1. Given a line segment on the $x$-axis, what's the boundary? How do we describe the segment mathematically. How do we describe the boundary mathematically? What does $\int_C \frac{df}{dx}dx = f(b)-f(a)$ say in terms of boundaries?
2. Given a wire, what's the boundary? How do we describe the wire mathematically. How do we describe the boundary mathematically?
3. Given a region $R$ in the plane, what's the boundary? How do we describe the region mathematically? How do we describe the boundary mathematically?
4. Given a solid $D$ in space, what's the boundary? How do we describe the solid mathematically? How do we describe the boundary mathematically?

Wires and Surfaces

1. What is the difference between $y=f(x)$ and $\vec r(x) = (x,f(x))$?
2. What is the difference between $z=f(x,y)$ and $\vec r(x,y) = (x,y,f(x,y))$?
3. How do we mathematically describe a wire?
4. How do we mathematically describe a surface?
5. How do we find the length $s$ of a wire? We add up little bits of length $ds$.
   • What does each part of $\int_C ds = \int_C |\frac{d\vec r}{dt}|dt$ mean?
6. How do we find the surface area $\sigma$ of a surface? We add up little bits of surface area $d\sigma$.
   • What does each part of $\iint_S d\sigma = \iint |\frac{\partial\vec r}{\partial u}\times\frac{\partial\vec r}{\partial v}|dudv$ mean?

Fundamental Theorems

In the work below, note that $\vec T$ is a unit tangent vector to a curve, while $\vec n$ is a unit vector that is normal to a boundary and pointing away from a region.

• $f(b)-f(a) = \int_a^b \frac{df}{dx} dx$ - Fundamental Theorem of Calculus
• $f(B)-f(A) = \int_C \vec \nabla f\cdot \vec T ds$ - Fundamental Theorem of Line Integrals
• $\int_C \vec F\cdot \vec T ds = \iint_R (\vec \nabla \times \vec F)\cdot (0,0,1)dA$ - Green's Theorem
• $\int_C \vec F\cdot \vec T ds = \iint_S (\vec \nabla \times \vec F)\cdot \vec n d\sigma$ - Stokes's Theorem
• $\int_C \vec F\cdot \vec n ds = \iint_R (\vec \nabla \cdot \vec F)dA$ - Green's theorem
• $\iint_R \vec F\cdot \vec n d\sigma = \iiint_D (\vec \nabla \cdot \vec F)dV$ - Divergence Theorem

Problem Set
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