Unit 9: Curl, Divergence, and Parametrized Surfaces
A paddlewheel that spins or sits still, a point that sources or sinks fluid, and the surfaces that flux gets measured through.
9 topics · 28 questions
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9.1 The Curl of a Vector Field
- How it worksHow does the combination of partial derivatives Qx - Py measure the local spinning tendency of a two-dimensional vector field?
- Worked exampleFind the curl of the vector field F(x, y) = <-y, x>.
- Common mistakeWhy doesn't a vector field that visually curves and swirls have to have nonzero curl at every point?
Your turn1 of 2
Find the curl (Qx - Py) of the vector field F(x, y) = <4·x·y2, 4·x2·y>, as a function of x and y.9.2 The Divergence of a Vector Field
- How it worksHow does adding up Px and Qy measure whether a vector field is expanding outward or collapsing inward at a point?
- Worked exampleFind the divergence of the vector field F(x, y) = <x, y> at the point (1, 1).
- Common mistakeWhy isn't a vector field with divergence zero automatically a field where nothing at all is happening?
- Going furtherWhat happens to the divergence at a point that is acting as a sink instead of a source?
Your turn1 of 2
Find the divergence (Px + Qy) of the vector field F(x, y) = <4·x3, 4·y3>, as a function of x and y.9.3 Parametrized Surfaces
- How it worksHow do two parameter functions u and v sweep out a two-dimensional surface sitting inside three-dimensional space?
- Common mistakeWhy isn't a parametrized surface's equation the same thing as the level-surface equation you would write for the same shape?
- Worked exampleFind a parametrization for the sphere of radius 3 centered at the origin using spherical angles as parameters.
Your turn1 of 2
The sphere of radius 4 is parametrized by r(u, v) = <4·sin(u)·cos(v), 4·sin(u)·sin(v), 4·cos(u)>. Find its x-component, as a function of u and v.9.4 The Normal Vector to a Parametrized Surface
- How it worksHow does taking the cross product of the two partial derivative vectors ru and rv produce a vector normal to the surface?
- Worked exampleFind the normal vector to the surface r(u, v) = <u, v, u2 + v2> at the point where u = 1, v = 0.
- Going furtherWhat happens to the surface's normal vector direction when the roles of the parameters u and v are swapped?
Your turn1 of 2
For r(u, v) = <u, v, 4·u2 + 4·v2>, find the x-component of the normal vector ru x rv, as a function of u.9.5 Surface Area of a Parametrized Surface
- How it worksHow does the magnitude of ru cross rv measure the local area-stretching factor of a parametrized surface?
- Worked exampleFind the surface area of the parametrized surface r(u, v) = <u, v, u + v> for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.
- Ties togetherHow does this surface-area formula generalize the earlier surface-area formula built from ordinary partial derivatives fx and fy?
Your turn1 of 2
Find the surface area of the parametrized surface r(u, v) = <u, v, 4·u + 4·v> for 0 ≤ u ≤ 4, 0 ≤ v ≤ 2.9.6 Flux Integrals Through a Surface
- How it worksHow does the dot product of a vector field with the surface's normal vector, integrated over the surface, add up total flux?
- Worked exampleFind the flux of F(x, y, z) = <0, 0, 1> through the parametrized surface r(u, v) = <u, v, 0> for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.
- Common mistakeWhy doesn't reversing a surface's chosen orientation leave a flux integral's value unchanged?
Your turn1 of 2
Find the flux of F(x, y, z) = <0, 0, 4> through the parametrized surface r(u, v) = <u, v, 0> for 0 ≤ u ≤ 3, 0 ≤ v ≤ 4.9.7 Orientation of Surfaces and Curves
- How it worksHow does choosing a consistent normal vector direction across an entire surface define that surface's orientation?
- Common mistakeWhy isn't every surface in space guaranteed to have two separate, consistent sides to choose an orientation from?
- Worked exampleFind the outward-orientation normal vector for the sphere x2 + y2 + z2 = 1 at the point (0, 0, 1).
Your turn1 of 2
The unit sphere is parametrized by r(u, v) = <sin(u)·cos(v), sin(u)·sin(v), cos(u)>. Its outward unit normal at each point equals r(u,v) itself. Find its x-component, as a function of u and v.9.8 Vector Fields in Fluid Flow: Sources, Sinks, and Rotation
- How it worksHow do curl and divergence, computed from the very same vector field, end up measuring two completely independent physical behaviors?
- Worked exampleFind both the curl and the divergence of the vector field F(x, y) = <x2, y2>, and describe the flow they predict.
- Ties togetherHow do curl and divergence together give a complete local picture of a fluid flow's rotation and expansion at every point?
Your turn1 of 2
Find the divergence of the vector field F(x, y) = <4·x2, 4·y2>, as a function of x and y.9.9 Surfaces of Revolution as Parametrized Surfaces
- How it worksHow do a profile curve and a rotation angle combine into the two parameters that describe a surface of revolution?
- Worked exampleFind a parametrization for the surface obtained by revolving the curve z = y2 around the z-axis.
- Going furtherWhat happens to the parametrization when the profile curve is revolved only partway around the axis instead of a full circle?
Your turn1 of 2
Revolving the profile curve z = 4·s2 (s the distance from the axis) around the z-axis gives Z(s, theta) = 4·s2. Write it as a function of s.