← Math 21a: Multivariable Calculus

Unit 7: Vector Fields and Line Integrals

An arrow at every point in the plane, and the integral that tracks how much a field pushes along a path through it.

11 topics · 36 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

7.1 Vector Fields

  1. How it worksHow do the two component functions P(x, y) and Q(x, y) assign a vector to every point in the plane?
  2. Worked exampleSketch the vector field F(x, y) = <-y, x> at the points (1, 0), (0, 1), and (-1, 0).
  3. Common mistakeWhy isn't a vector field just a single vector attached to the whole plane at once?
Your turn1 of 2
Find the magnitude of the vector field F(x, y) = <-4·y, 4·x> at the point (x, y), as a function of x and y.

7.2 Gradient Fields and Potential Functions

  1. How it worksHow does taking the gradient of a scalar function f turn it into a vector field pointing toward steepest increase everywhere?
  2. Worked exampleFind the gradient field of f(x, y) = x2 y.
  3. Going furtherWhat happens to a gradient field at a point where the underlying scalar function has a critical point?
Your turn1 of 2
Find the x-component of the gradient field of f(x, y) = 4·x2·y, as a function of x and y.

7.3 Line Integrals of Scalar Functions

  1. How it worksHow does parametrizing a curve and multiplying by the speed |r'(t)| turn a scalar line integral into an ordinary single integral?
  2. Worked exampleFind the line integral of f(x, y) = x + y along the curve r(t) = <t, t2> for t from 0 to 1.
  3. Common mistakeWhy doesn't a scalar line integral change value when you travel along the curve in the opposite direction?
Your turn1 of 2
Find the line integral of f(x, y) = 4·(x+y) along the straight path r(t) = <3·t, 4·t> for t from 0 to 2.

7.4 Line Integrals of Vector Fields

  1. How it worksHow does taking the dot product of a vector field with the velocity vector along a path add up the field's push along it?
  2. Common mistakeWhy does a line integral of a vector field flip sign the instant the direction of travel along the curve reverses?
  3. Worked exampleFind the line integral of F(x, y) = <y, x> along the quarter circle r(t) = <cos(t), sin(t)> for t from 0 to π/2.
Your turn1 of 2
Find the line integral of F(x, y) = <4·y, 4·x> along r(t) = <cos(t), sin(t)> for t from 0 to 1.08.

7.5 Work Done by a Force Field

  1. How it worksHow does the line integral of a force field along a path compute the total work done moving an object through that path?
  2. Worked exampleFind the work done by the force field F(x, y) = <x, y> moving a particle along the straight line from (0, 0) to (1, 1).
  3. Ties togetherHow does computing work as a line integral of a force field connect back to work-energy ideas from single-variable physics?
Your turn1 of 2
Find the work done by the force field F(x, y) = <4·x, 4·y> moving a particle along the straight line from (0, 0) to (6, 2).

7.6 The Fundamental Theorem of Line Integrals

  1. How it worksHow does the fundamental theorem of line integrals turn integrating a gradient field into evaluating a potential function at the endpoints?
  2. Worked exampleFind the line integral of the gradient of f(x, y) = x2 y from (0, 0) to (1, 2) using the fundamental theorem of line integrals.
  3. Common mistakeWhy doesn't the fundamental theorem of line integrals apply to a vector field that isn't itself a gradient field?
  4. Going furtherWhat happens to a line integral over a closed loop when the vector field being integrated is a gradient field?
Your turn1 of 2
Using the fundamental theorem of line integrals, find the line integral of the gradient of f(x, y) = 4·x2·y from (0, 0) to (x, y), as a function of x and y.

7.7 Conservative Vector Fields

  1. How it worksHow does path independence of a line integral prove that a vector field must be the gradient of some potential function?
  2. Common mistakeWhy isn't every vector field that looks smooth and well-behaved automatically conservative?
  3. Worked exampleFind a potential function f for the vector field F(x, y) = <2xy, x2>, confirming F is conservative.
Your turn1 of 2
Find a potential function f (with f(0,0)=0) for the vector field F(x, y) = <4·2·x·y, 4·x2>, as a function of x and y.

7.8 Testing for Conservative Fields: the Curl Test

  1. How it worksHow does the equality of the partial derivatives Py and Qx confirm that a two-dimensional vector field is conservative?
  2. Worked exampleShow whether F(x, y) = <2xy, x2 + 3y2> is conservative, using the curl test.
  3. Going furtherWhat happens to the curl test's conclusion when the domain of the vector field has a hole punched out of its center?
  4. Common mistakeWhy doesn't a vector field with zero curl automatically have to be conservative on every possible domain?
Your turn1 of 2
For F(x, y) = <6·x·y, x2 + 4·y2>, with P = 6·x·y and Q = x2 + 4·y2, write the curl-test quantity Qx - Py, as a function of x.

7.9 Path Independence

  1. How it worksHow does splitting a closed loop into two paths with opposite direction show that path independence and zero circulation are the same statement?
  2. Worked exampleShow that the line integral of F(x, y) = <2xy, x2> is the same along the straight path and the parabolic path from (0, 0) to (1, 1).
  3. Ties togetherHow does path independence tie together the fundamental theorem of line integrals and the definition of a conservative field?
Your turn1 of 2
F(x, y) = <7·x·y, (7/2)·x2> is conservative with potential f(x,y)=(7/2)·x2·y. Using path independence, find the line integral of F from (0, 0) to (x, y), as a function of x and y.

7.10 Line Integrals Around Closed Curves

  1. How it worksHow does the closed-curve line integral of a vector field measure the net circulation of that field around the loop?
  2. Worked exampleFind the line integral of F(x, y) = <-y, x> around the unit circle traversed counterclockwise.
  3. Going furtherWhat happens to a closed-curve line integral when the direction of travel around the loop reverses from counterclockwise to clockwise?
Your turn1 of 2
Find the line integral of F(x, y) = <-4·y, 4·x> around the unit circle traversed counterclockwise.

7.11 Vector Fields and Flow Lines

  1. How it worksHow does solving for a curve whose velocity always matches the vector field at that point produce a flow line?
  2. Worked exampleFind the flow line of the vector field F(x, y) = <1, x> that passes through the origin.
  3. Common mistakeWhy isn't a flow line of a vector field the same thing as a level curve of some function?
  4. Going furtherWhat happens to a family of flow lines near a point where the vector field itself equals the zero vector?
Your turn1 of 2
Find the flow line of the vector field F(x, y) = <4, 4·x> that passes through the origin, as y as a function of x.