← Calculus III (Multivariable)

Unit 5: Vector Calculus

A vector field carries a dropped-in particle, curl spins a paddlewheel, and three theorems show the boundary already knows what the interior is doing.

26 topics · 98 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.

6.1 Vector Fields and Their Flow Lines

  1. How it worksHow does dropping a particle into a vector field and letting it ride the arrows trace out a flow line?
  2. Worked exampleSketch the vector field F(x, y) = (-y, x) and identify its flow lines.
  3. Common mistakeWhy isn't a vector field just a scattered set of individual force vectors with nothing connecting them?
Your turn1 of 2
Find the magnitude of the vector field F(x, y) = (-4y, 4x), as a function of x and y.

6.1 Gradient Fields and Potential Functions

  1. How it worksHow does taking the gradient of a scalar function turn it into a vector field pointing toward increase?
  2. Worked exampleFind the gradient field of f(x, y, z) = x2 y + yz2.
  3. Going furtherWhat happens to a gradient field's arrows near a local maximum of its underlying scalar function?
  4. Worked exampleFind a potential function for the gradient field F(x, y) = (2x, 2y) and identify its level curves.
Your turn1 of 2
Find the x-component of the gradient field of f(x, y, z) = 4x2y + yz2.

6.2 Line Integrals of a Scalar Function Along a Curve

  1. How it worksHow does parametrizing a curve turn a line integral of a scalar function into an ordinary single integral?
  2. Worked exampleFind the line integral of f(x, y) = xy along the curve r(t) = (t, t2) for t from 0 to 1.
  3. Common mistakeWhy doesn't a scalar line integral care which direction you travel along the curve?
  4. Worked exampleFind the line integral of f(x, y) = x2 along the line segment from (0, 0) to (1, 1).
Your turn1 of 2
Find the line integral of f(x, y) = 4x2 along the line segment from (0, 0) to (4, 4).

6.2 Line Integrals with Respect to Arc Length

  1. How it worksHow does a line integral with respect to arc length differ from one taken with respect to dx or dy alone?
  2. Worked exampleFind the line integral of f(x,y) = x + y with respect to arc length along the segment from (0,0) to (3,4).
  3. Common mistakeWhy doesn't reversing direction along a curve change a line integral taken with respect to arc length, even though it flips a vector line integral?
  4. Going furtherWhat happens to a line integral with respect to arc length once the density function is constant along the whole curve?
Your turn1 of 2
Find the line integral of f(x,y) = 4x + 4y with respect to arc length along the segment from (0,0) to (6, 2).

6.2 Line Integrals of a Vector Field: Work Done Along a Path

  1. How it worksHow does dotting a vector field with a curve's tangent vector at every point add up to total work?
  2. Worked exampleFind the work done by F(x, y) = (y, x) moving a particle along r(t) = (t, t2) from t = 0 to t = 1.
  3. Worked exampleFind the work done by F(x, y) = (-y, x) moving a particle once around the unit circle counterclockwise.
  4. Common mistakeWhy does reversing the direction of travel along a curve flip the sign of a vector field's line integral?
Your turn1 of 2
Find the work done by F(x, y) = (4y, 4x) moving a particle along r(t) = (t, t2) from t = 0 to t = 2.05.

6.2 Circulation and Flux of a Vector Field Along a Curve

  1. How it worksHow does dotting a vector field with a curve's normal instead of its tangent measure flux instead of circulation?
  2. Worked exampleFind the flux of F(x, y) = (x, y) across the unit circle traversed counterclockwise.
  3. Ties togetherHow does flux across a plane curve foreshadow the flux integrals used later for surfaces in three dimensions?
Your turn1 of 2
Find the flux of F(x, y) = (4x, 4y) across the circle of radius 4 centered at the origin, traversed counterclockwise.

6.3 The Fundamental Theorem for Line Integrals

  1. How it worksHow does the fundamental theorem for line integrals turn work into a difference of potential function values?
  2. Worked exampleFind the work done by F = gradient of f, where f(x,y) = x2 y, moving from (0,0) to (1,2).
  3. Common mistakeWhy doesn't this endpoint shortcut work for a vector field that isn't the gradient of some scalar function?
  4. Worked exampleFind the work done by F = gradient of f, where f(x,y,z) = xyz, moving from (1,1,1) to (2,3,1).
Your turn1 of 2
Find the work done by F = gradient of f, where f(x,y) = 4x2y, moving from (0,0) to (1, 4).

6.3 Path Independence of Line Integrals

  1. How it worksHow does path independence for every possible path between two points turn out to be equivalent to a field being conservative?
  2. Worked exampleShow that the line integral of F(x,y) = (2xy, x2) gives the same value along two different paths from (0,0) to (1,1).
  3. Ties togetherHow does path independence connect the fundamental theorem for line integrals back to the ordinary fundamental theorem of calculus?
Your turn1 of 2
By path independence, find the line integral of F(x, y) = (24xy, 4x2) from (0,0) to (1, 4).

6.3 Testing Whether a Vector Field Is Conservative

  1. How it worksHow does checking that a two-dimensional vector field's cross-partials match test whether it's conservative?
  2. Worked exampleFind out whether F(x, y) = (2xy, x2 + 3y2) is conservative.
  3. Worked exampleFind out whether F(x, y) = (y, -x) is conservative.
  4. Common mistakeWhy doesn't a vector field with all positive components automatically fail to be conservative?
Your turn1 of 2
For F(x, y) = (6xy, x2 + 4y2), with P = 6xy and Q = x2 + 4y2, write the conservative-field test quantity partial Q partial x minus partial P partial y, as a function of x.

6.3 Finding a Potential Function

  1. How it worksHow does integrating one component and comparing it with the other recover a vector field's potential function?
  2. Worked exampleFind a potential function f for F(x, y) = (2xy + 1, x2 - 3y2).
  3. Going furtherWhat happens to the potential function you recover once a vector field is conservative on a domain with a hole cut out of it?
  4. Worked exampleFind a potential function for F(x, y, z) = (yz, xz, xy).
Your turn1 of 2
Find a potential function f (with f(0,0)=0) of F(x, y) = (2xy + 4, x2 - 4y2), as a function of x and y.

6.3 Why Curl-Free Does Not Always Mean Conservative

  1. Common mistakeWhy isn't a two-dimensional vector field automatically conservative just because its curl is zero everywhere it's defined?
  2. How it worksHow does the vector field (-y, x)/(x2+y2) manage to be curl-free everywhere except the missing origin, yet not be conservative?
  3. Worked exampleFind the line integral of F(x,y) = (-y, x)/(x2+y2) around the unit circle and show it isn't zero despite F being curl-free.
  4. Ties togetherHow does the domain needing to be simply connected explain why a hole in the plane breaks the curl test?
Your turn1 of 2
Find the line integral of F(x, y) = (-4y, 4x)/(x2+y2) around the circle of radius 6 centered at the origin, traversed counterclockwise.

6.4 Green's Theorem: Circulation Form

  1. How it worksHow do the tiny circulations of neighboring interior cells cancel along shared edges, leaving only the boundary's circulation?
  2. Worked exampleFind the line integral of (y, x2) around the boundary of the square [0,1] x [0,1], using Green's theorem.
  3. Common mistakeWhy doesn't Green's theorem apply to a curve that isn't simple and closed?
  4. Worked exampleFind the circulation of F(x,y) = (-y, x) around the boundary of the triangle with vertices (0,0), (1,0), (0,1), using Green's theorem.
Your turn1 of 2
Find the line integral of (4y, x2) around the boundary of the square [0,4] x [0,4], using Green's theorem.

6.4 Green's Theorem: Flux Form

  1. How it worksHow does rewriting Green's theorem in terms of the divergence turn circulation into flux?
  2. Worked exampleFind the flux of F(x, y) = (x3, y3) across the boundary of the unit circle, using the flux form of Green's theorem.
  3. Ties togetherHow do the circulation and flux forms of Green's theorem relate to curl and divergence respectively?
Your turn1 of 2
Find the flux of F(x, y) = (4x3, 4y3) across the boundary of the circle of radius 4 centered at the origin, using the flux form of Green's theorem.

6.4 Using Green's Theorem to Compute Area

  1. How it worksHow does choosing a clever vector field turn Green's theorem into a pure area formula?
  2. Worked exampleFind the area enclosed by the ellipse x = 3cos t, y = 2 sin t, using Green's theorem.
  3. Going furtherWhat happens to the area formula from Green's theorem once the curve is traversed clockwise instead of counterclockwise?
  4. Worked exampleFind the area enclosed by the astroid x = cos3(t), y = sin3(t), using Green's theorem.
Your turn1 of 2
Find the area enclosed by the ellipse x = 6cos t, y = 6sin t, using Green's theorem.

6.4 Green's Theorem on Regions with Holes

  1. How it worksHow does adding a cut between the two boundary curves turn an annular region into one simple closed curve Green's theorem can handle?
  2. Worked exampleFind the circulation of F(x,y) = (-y,x)/(x2+y2) around the boundary of the annulus 1 ≤ x2+y2 ≤ 4, using Green's theorem.
  3. Ties togetherHow does this hole-in-the-region trick explain why the curl-free field from before still has nonzero circulation around the origin?
Your turn1 of 2
For F(x, y) = (-4y, 4x)/(x2+y2), the circulation around a circle of radius 6 centered at the origin (counterclockwise) is the same for every such circle, since F is curl-free away from the origin. Find that common value.

6.5 Curl of a Vector Field and Rotation

  1. How it worksHow does the curl formula measure the net rotation of a vector field around a point?
  2. Worked exampleFind the curl of F(x, y, z) = (y, -x, 0) and confirm it matches a rigid rotation.
  3. Worked exampleFind the curl of F(x, y, z) = (yz, xz, xy).
  4. Common mistakeWhy doesn't a vector field with long, sweeping, curved arrows automatically have nonzero curl?
Your turn1 of 2
Find the z-component of the curl of F(x, y, z) = (ay, -ax, 0), as a function of a.

6.5 Divergence of a Vector Field: Sources and Sinks

  1. How it worksHow does the divergence formula measure whether a vector field is expanding or contracting around a point?
  2. Worked exampleFind the divergence of F(x, y, z) = (x2, y2, z2) at the point (1, 1, 1).
  3. Common mistakeWhy doesn't a vector field with zero divergence everywhere have to be the zero field itself?
  4. Worked exampleFind the divergence of F(x, y, z) = (x, y, z) and interpret it as a constant outward source strength.
Your turn1 of 2
Find the divergence of F(x, y, z) = (4x2, 4y2, 4z2), as a function of x, y, and z.

6.5 Combining Gradient, Divergence, and Curl with the Del Operator

  1. How it worksHow does treating del as a vector of partial derivative operators unify the formulas for gradient, divergence, and curl?
  2. Worked exampleFind div(curl F) for F(x, y, z) = (xy, yz, xz) and confirm it equals zero.
  3. Common mistakeWhy isn't del an actual vector the way i, j, and k are?
  4. Ties togetherHow does the identity div(curl F) = 0 explain why every curl field is automatically divergence-free?
Your turn1 of 2
Find div(curl F) for F(x, y, z) = (4xy, 4yz, 4xz).

6.6 Parametric Surfaces in Space

  1. How it worksHow do two parameters u and v sweep out a surface through a vector-valued function r(u, v)?
  2. Worked exampleFind a parametrization for the sphere of radius 3 centered at the origin.
  3. Worked exampleFind a parametrization for the part of the paraboloid z = x2 + y2 below z = 4.
  4. Going furtherWhat happens to a parametric surface's shape once one of the two parameters is restricted to a smaller interval?
Your turn1 of 2
Using the parametrization r(u,v) = (6sin(u)cos(v), 6sin(u)sin(v), 6cos(u)) for the sphere of radius 6, write the z-coordinate as a function of u.

6.6 Surface Area of a Parametric Surface

  1. How it worksHow does the cross product of the two partial derivatives ru and rv measure a parametric surface's local area-stretch?
  2. Worked exampleFind the surface area of the sphere of radius 2, using its parametrization in spherical angles.
  3. Ties togetherHow does this cross-product area element generalize the same stretch idea the Jacobian measures for a coordinate change?
Your turn1 of 2
Find the surface area of a sphere, using its parametrization in spherical angles, as a function of its radius R.

6.6 Surface Integrals of a Scalar Function

  1. How it worksHow does the surface area element dS turn a surface integral of a scalar function into a double integral over parameters?
  2. Worked exampleFind the surface integral of f(x, y, z) = z over the part of the plane x + y + z = 1 in the first octant.
  3. Going furtherWhat happens to a scalar surface integral once the density function is constant across the whole surface?
  4. Worked exampleFind the surface integral of f(x, y, z) = 1 over the hemisphere x2 + y2 + z2 = 4, z ≥ 0, to recover its surface area.
Your turn1 of 2
Find the surface integral of f(x, y, z) = 4z over the part of the plane x + y + z = 4 in the first octant.

6.6 Orientation of a Surface

  1. How it worksHow does a continuously varying choice of unit normal vector define an orientation for a surface?
  2. Common mistakeWhy doesn't every surface in space actually have two distinct sides to choose between?
  3. Worked exampleFind the outward unit normal vector to the sphere x2 + y2 + z2 = 4 at the point (2, 0, 0).
  4. Going furtherWhat happens to a flux integral's sign once a closed surface's orientation is switched from outward to inward?
Your turn1 of 2
A point (x, y, z) lies on a sphere centered at the origin. Write the x-component of the outward unit normal vector to the sphere at that point, as a function of x, y, and z.

6.6 Surface Integrals of a Vector Field: Flux

  1. How it worksHow does dotting a vector field with a surface's unit normal at every point add up to total flux?
  2. Worked exampleFind the flux of F(x, y, z) = (0, 0, z) upward through the disk x2+y2 ≤ 1 in the plane z = 3.
  3. Worked exampleFind the flux of F(x, y, z) = (x, y, z) outward through the sphere of radius 1 centered at the origin.
  4. Common mistakeWhy does flipping a surface's chosen orientation flip the sign of its flux integral?
Your turn1 of 2
Find the flux of F(x, y, z) = (0, 0, 4z) upward through the disk x2+y2 ≤ 62 in the plane z = 6.

6.7 Stokes' Theorem

  1. How it worksHow do the tiny circulations of neighboring surface patches cancel along shared edges, leaving only the boundary curve's circulation?
  2. Worked exampleFind the line integral of F(x,y,z) = (y, -x, z) around the boundary of the disk x2+y2 ≤ 1 at z = 0, using Stokes' theorem.
  3. Ties togetherHow does Stokes' theorem reduce to Green's theorem once the surface is flat and lies in the xy-plane?
Your turn1 of 2
Find the line integral of F(x,y,z) = (4y, -4x, 4z) around the boundary of the disk x2+y2 ≤ 62 at z = 0, using Stokes' theorem.

6.8 The Divergence Theorem

  1. How it worksHow do the tiny fluxes of neighboring interior cells cancel across shared faces, leaving only the outer boundary's flux?
  2. Worked exampleFind the flux of F(x, y, z) = (x, y, z) outward through the sphere of radius 2, using the divergence theorem.
  3. Worked exampleFind the flux of F(x, y, z) = (x2, y2, z2) through the boundary of the cube [0,1]3, using the divergence theorem.
  4. Common mistakeWhy doesn't the divergence theorem apply once a vector field has a singularity somewhere inside the enclosed solid?
Your turn1 of 2
Find the flux of F(x, y, z) = (4x, 4y, 4z) outward through the sphere of radius 4, using the divergence theorem.

6.8 Unifying the Fundamental Theorem, Green's, Stokes', and Divergence Theorems

  1. Ties togetherHow do the line integral, Green's, Stokes', and divergence theorems all generalize the one-variable fundamental theorem?
  2. How it worksHow does each of these theorems trade an integral of a derivative over a region for an integral of the original quantity over its boundary?
  3. Going furtherWhat changes about this boundary-versus-interior pattern once you move from a two-dimensional region to a three-dimensional solid?
  4. Worked exampleShow that the flux of curl F through two different surfaces sharing the same boundary curve gives the same value for F(x,y,z) = (y,-x,0).
  5. Common mistakeWhy doesn't Green's theorem require a vector field to be conservative in order to apply?
Your turn1 of 2
Why doesn't Green's theorem require the vector field F=(P,Q) to be conservative in order to apply?