Vector Calculus

Units

1. 1

   Derivatives as matrices

   4 lessons

   1. The derivative matrixWrite the derivative of a map F from Rⁿ to Rᵐ at a point as the m × n matrix of partial derivatives, and use it as the best linear approximation F(a + h) ≈ F(a) + DF(a)h.
   2. The chain rule as matrix multiplicationCompute the derivative matrix of a composition as the product D(f ∘ g)(a) = Df(g(a)) Dg(a), check that the sizes fit, and recover the one-variable chain rule and the gradient test for a level curve.
   3. Taylor's theorem in several variablesWrite the second-order Taylor polynomial f(a) + ∇f(a)·h + ½hᵀHh with the Hessian H, see it as the quadratic surface that hugs z = f(x, y) at a point, and say how fast its error shrinks.
   4. The Hessian and the second derivative testClassify a critical point of f(x, y) by the eigenvalues of the Hessian (both positive: minimum, both negative: maximum, opposite signs: saddle), read the 2 × 2 determinant test from them, and say when the test fails.
2. 2

   Extrema and Lagrange multipliers

   3 lessons

   1. Local maxima, minima and saddle pointsFind the critical points of f(x, y) by solving f_x = 0 and f_y = 0 together, and classify each as a local maximum, a local minimum or a saddle point with the second derivative test D = f_xx f_yy − (f_xy)², including what to do when D = 0.
   2. Absolute extrema on a closed regionFind the largest and smallest values of f(x, y) on a closed and bounded region by listing the interior critical points and the one-variable extremes on each edge, comparing all values, and extend the idea to a box-volume word problem.
   3. Lagrange multipliersFind the largest and smallest values of f on a constraint curve g(x, y) = k by solving ∇f = λ∇g together with g = k, understand why the gradients are parallel, compare the candidate values, and use the same method for a function of three variables.
3. 3

   Coordinates and change of variables

   4 lessons

   1. Double integrals in polar coordinatesDescribe a disk, ring or sector by constant polar limits, see from a polar cell why dA = r dr dθ, convert a double integral to polar form, and use it for volumes, regions between curves and the Gaussian integral.
   2. Cylindrical coordinatesName points of space by (r, θ, z), convert both ways, see why the volume element carries a factor r, and convert and evaluate triple integrals over solids with a round axis.
   3. Spherical coordinatesName points of space by (ρ, φ, θ) with φ measured from the positive z-axis, convert both ways, see why the volume element is ρ² sin φ dρ dφ dθ, and integrate over balls and cones.
   4. Change of variables and the JacobianUse a transformation to turn a tilted or curved region into a simple one, find the Jacobian as the local area stretch factor, and write ∬_R f dA as ∬_S f |J| du dv, including the ellipse, polar and spherical cases.
4. 4

   Vector fields and line integrals

   5 lessons

   1. Vector fieldsUnderstand a vector field as an arrow attached to every point, read and evaluate F = ⟨P, Q⟩ for the swirl ⟨−y, x⟩, follow flow lines, meet gradient fields and show the swirl is not one, and extend to fields in space such as the inverse-square field.
   2. Line integrals of scalar functionsAdd up a scalar function f along a curve C as ∫_C f ds, compute it with ds = |r′(t)| dt, see that it ignores the pace and the direction of travel, and extend it to pieces, closed curves and wires in space.
   3. Line integrals of vector fieldsCompute the work ∫_C F · dr = ∫ F(r(t)) · r′(t) dt of a force field along a path, see how orientation changes the sign and how the route changes the answer, and use the form P dx + Q dy and the circulation round a closed curve.
   4. The fundamental theorem for line integralsProve and use ∫_C ∇f · dr = f(B) − f(A), see that the work of a gradient field depends only on the end points and is zero round any closed curve, and contrast it with the swirl ⟨−y, x⟩, which is not a gradient field.
   5. Conservative fields and potential functionsDecide whether a vector field is the gradient of a function, using the cross partials, find that function, and use it to evaluate work from the two endpoints alone, while knowing why the test needs a region with no holes.
5. 5

   Green's theorem, curl and divergence

   2 lessons

   1. Green's theoremTurn the circulation of a field around a closed curve in the plane into a double integral over the region inside, keep the orientation and the conditions right, and use the theorem to evaluate hard line integrals and to find areas.
   2. Curl and divergenceMeasure how a field spins and how it spreads out at a point, write both with the del operator, use the two identities that follow, and meet Green's theorem in its circulation and flux forms.
6. 6

   Surfaces and surface integrals

   3 lessons

   1. Parametric surfaces and their areasDescribe a curved surface with two parameters, r(u, v), read its grid curves, find tangent vectors, the normal and the tangent plane at a point, and compute its area with |r_u × r_v|, including the sphere, graphs and surfaces of revolution.
   2. Surface integrals of scalar functionsAdd a quantity such as density over a curved surface with ∬_S f dS = ∬_D f(r(u, v)) |r_u × r_v| dA, use the graph form dS = √(1 + g_x² + g_y²) dA, and find the mass and the average value of a thin shell.
   3. Surface integrals of vector fields (flux)Measure how much a flowing field crosses a surface per second with ∬_S F · n dS = ∬_D F(r(u, v)) · (r_u × r_v) dA, choose and respect the orientation of the surface, and find the flux of F = ⟨x, y, z⟩ through a sphere.
7. 7

   The theorems of Stokes and Gauss

   3 lessons

   1. Stokes' theoremState and use Stokes' theorem, that the circulation of a field round the rim of a surface equals the flux of its curl through the surface, orient the rim and the surface with the right-hand rule, choose whichever surface is easiest, and see Green's theorem as the flat case.
   2. The divergence theoremState and use the divergence theorem, that the flux of a field out of a closed surface equals the integral of its divergence over the solid inside, read divergence as a source strength, and close an open surface to turn a hard flux into an easy triple integral.
   3. The great theorems side by sideSee the fundamental theorem of calculus, the fundamental theorem for line integrals, Green's theorem, Stokes' theorem and the divergence theorem as one pattern, choose the right one by what is integrated over what, and check an answer by a second route.
8. 8

   Vector calculus at work

   3 lessons

   1. Vector identities and potentialsProve curl(∇f) = 0 and div(curl F) = 0 from equal mixed partials, and use them to decide when a field has a potential or a vector potential.
   2. Conservation laws and the continuity equationDerive ∂ρ/∂t + div(ρv) = 0 from the divergence theorem and read it as a balance: mass inside a region changes only by flux through its boundary.
   3. Inverse-square fields and Gauss's lawShow that F = x/|x|³ has div F = 0 away from the origin yet flux 4π out of every sphere about it, and use the divergence theorem on a region with a hole to explain why.