← Math 21a: Multivariable Calculus

Unit 2: Curves and Surfaces in Space

Curves traced by a moving point, and surfaces sliced open by planes until every quadric shape gives up its structure.

14 topics · 49 questions

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2.1 Parametrized Curves

  1. How it worksHow do three component functions x(t), y(t), and z(t) trace out a curve as t increases?
  2. Worked exampleFind the point on the curve r(t) = <cos(t), sin(t), t> at t = π.
  3. Common mistakeWhy isn't a parametrized curve the same thing as the plain set of points it passes through?
Your turn1 of 2
A curve is r(t) = <4·cos(t), 4·sin(t), 4·t>. Write its x-coordinate as a function of t.

2.2 Velocity and Acceleration Vectors

  1. How it worksHow does differentiating a position vector component by component produce the velocity vector?
  2. Worked exampleFind the velocity and acceleration of r(t) = <t2, t3, t> at t = 1.
  3. Common mistakeWhy doesn't traveling at constant speed mean the acceleration vector is zero?
  4. Going furtherWhat happens to the velocity vector at a point where a curve has a sharp corner?
Your turn1 of 2
For r(t) = <4·t2, 4·t3, 6·t>, find the y-component of the velocity vector r'(t), as a function of t.

2.3 Arc Length of a Space Curve

  1. How it worksHow does integrating the speed |r'(t)| over time add up the length of a curving path?
  2. Worked exampleFind the arc length of r(t) = <3cos(t), 3sin(t), 4t> for t from 0 to 2pi.
  3. Going furtherWhat happens to the arc length integral when a curve is traced at twice the speed but follows the same path?
Your turn1 of 2
Find the speed |r'(t)| for r(t) = <4·t, 4·t2, 0>, as a function of t.

2.4 The Unit Tangent Vector and Curvature

  1. How it worksHow does dividing the velocity vector by its own length produce a unit tangent vector that tracks direction alone?
  2. How it worksHow does the curvature formula measure how fast the unit tangent vector swings as you move along a curve?
  3. Worked exampleFind the curvature of the circle r(t) = <cos(t), sin(t)> of radius 1.
  4. Common mistakeWhy does a straight line always have zero curvature no matter how it gets parametrized?
Your turn1 of 2
Find the curvature of r(t) = <t, 4·t3, 0>, as a function of t.

2.5 Functions of Two and Three Variables

  1. How it worksHow does plugging in an x-value and a y-value produce the height z of a surface at that point?
  2. Worked exampleFind f(2, 3) for f(x, y) = x2 - 2xy + y2.
  3. Going furtherWhat happens to a function of two variables when you freeze one variable and let only the other move?
Your turn1 of 2
Find f(5, -3) for f(x, y) = 4x2 - 4xy + y2.

2.6 Level Curves and Contour Maps

  1. How it worksHow does slicing a surface with a horizontal plane at height c produce that surface's level curve at c?
  2. Worked exampleFind and describe the level curve of f(x, y) = x2 + y2 at the value c = 9.
  3. Common mistakeWhy doesn't a level curve by itself tell you how steep the surface is at that height?
  4. Going furtherWhat happens to the spacing between level curves where a surface is very steep versus nearly flat?
Your turn1 of 2
The level curve of f(x, y) = 2x2 + y2 at height c = 16 can be solved for y (positive branch). Write that y as a function of x.

2.7 Level Surfaces

  1. How it worksHow does setting f(x, y, z) equal to a constant carve a level surface out of three-dimensional space?
  2. Worked exampleFind the level surface of f(x, y, z) = x2 + y2 + z2 at the value 25, and identify its shape.
  3. Ties togetherHow does a level surface of a three-variable function play the same role that a level curve plays for a two-variable function?
Your turn1 of 2
Find the radius of the level surface x2 + y2 + z2 = 21.

2.8 Cylinders and Traces

  1. How it worksHow does slicing a surface with planes parallel to each coordinate plane reveal its traces?
  2. Worked exampleFind and describe the traces of the surface z = x2 in the planes x = 0, y = 0, and z = 4.
  3. Common mistakeWhy isn't the surface x2 + y2 = 4 just a circle floating in space?
Your turn1 of 2
The cylinder x2 + y2 = 21 intersects every horizontal plane in a circle. Find that circle's radius.

2.9 Quadric Surfaces: Ellipsoids and Spheres

  1. How it worksHow do the traces of x2/a2 + y2/b2 + z2/c2 = 1 in each coordinate plane reveal it as an ellipsoid?
  2. Worked exampleFind the trace of the ellipsoid x2/4 + y2/9 + z2 = 1 in the plane z = 0.
  3. Going furtherWhat happens to an ellipsoid's shape as one of its three axis lengths grows without bound?
Your turn1 of 2
Find the area enclosed by the elliptical trace of the ellipsoid x2/72 + y2/62 + z2/82 = 1 in the plane z = 0.

2.10 Quadric Surfaces: Paraboloids and Cones

  1. How it worksHow do circular level curves at every height stack up into the surface z = x2 + y2?
  2. Worked exampleFind the level curve of the paraboloid z = x2 + y2 at height z = 4.
  3. Common mistakeWhy doesn't a cone have the same cross-sectional size no matter how far up you slice it?
  4. Going furtherWhat changes about the level curves as z increases for z = x2 + y2 versus for z2 = x2 + y2?
Your turn1 of 2
Find the radius of the circular level curve of the paraboloid z = 4(x2+y2) at height z = 20.

2.11 Quadric Surfaces: Hyperboloids and Saddles

  1. How it worksHow does flipping one sign in x2 + y2 - z2 = 1 turn an ellipsoid's formula into a hyperboloid of one sheet?
  2. Common mistakeWhy doesn't a saddle surface like z = x2 - y2 have a maximum or a minimum sitting at its center point?
  3. Worked exampleFind the traces of the hyperboloid x2 + y2 - z2 = 1 in the plane z = 0 and in the plane x = 0.
  4. Going furtherWhat changes about the surface x2 + y2 - z2 = 1 when the right-hand side flips from 1 to -1?
Your turn1 of 2
The hyperboloid x2 + y2/62 - z2/62 = 1 has a trace in the plane x = 0 given by y2/62 - z2/62 = 1. Write its y-value (positive branch) as a function of z.

2.12 Cylindrical and Spherical Coordinates as Surfaces

  1. How it worksHow do r = constant, theta = constant, and z = constant each carve out a different simple surface in cylindrical coordinates?
  2. Worked exampleFind the cylindrical-coordinate equation of the cone z = square root of x2 + y2.
  3. How it worksHow does rho = constant in spherical coordinates carve out a sphere centered at the origin?
  4. Going furtherWhat happens to the equation of a plane through the origin when it gets rewritten in spherical coordinates?
Your turn1 of 2
The cone z = 4*√(x2+y2), converted to cylindrical coordinates, becomes z = 4·r. Write z as a function of r.

2.13 Sketching Surfaces by Combining Traces

  1. How it worksHow do the traces in the xy-, xz-, and yz-planes combine to reveal the full three-dimensional shape of a quadric surface?
  2. Worked exampleSketch the surface z = x2 + 4y2 by finding and combining its traces in the planes x = 0, y = 0, and z = 4.
  3. Ties togetherHow does combining traces to sketch a surface identify which quadric type, like an ellipsoid or hyperboloid, it is?
Your turn1 of 2
The surface z = x2 + 4y2 has a trace in the plane z = h that is an ellipse. Write its y-intercept (positive branch) as a function of h.

2.14 Famous Parametrized Curves: Helices and Cycloids

  1. How it worksHow do a circular motion and a steady vertical climb combine into the parametrization of a helix?
  2. Worked exampleFind the position at t = π for the helix r(t) = <cos(t), sin(t), t>, and describe how it differs from a plain circle.
  3. Worked exampleFind the parametrization of a cycloid traced by a point on a wheel of radius 1 rolling along the x-axis.
  4. Going furtherWhat happens to a cycloid's shape when the point being tracked moves from the rim of the wheel to somewhere inside it?
Your turn1 of 2
Write the x-coordinate of the cycloid x(t) = 4·(t - sin(t)), traced by a point on a wheel of radius 4 rolling along the x-axis, as a function of t.