Calculus 3 (multivariable calculus), part 2 of 2

Revisit Riemann integrals, exploring properties like limit flipping, assigned area, linearity, additivity, and piecewise definitions, and relate these to applications such as area, mass, and volumes.

Explore geometric reasoning to estimate Riemann integrals by inspection using area pictures, rectangles, squares, triangles, half discs, and apply odd-function symmetry to cancel from minus a to a.

Explore odd functions in two variables, defined by f(-x,y) = -f(x,y) and f(x,-y) = -f(x,y), revealing origin symmetry, zeros on the axes, and estimation of integrals by inspection.

solve an integration by inspection problem on a symmetric domain d defined by |x|+|y| ≤ 1, showing the integral equals -2π by oddness and area.

Apply geometry and symmetry to the double integral of x plus three over the half-disk domain; the x term cancels, giving three times the half-disk area, six pi.

Describe x simple and y simple domains, and how axis-parallel rectangles and inequalities define them for Fubini's theorem and double integration.

Apply Fubini's theorem to a double integral on an x- or y-simple domain, determine the optimal order of integration for (x/y) e^y, and compute the result.

Solve the last problem in Fubini's theorem session, computing the double integral of log x over the region between the line 2x+2y=5 and the hyperbola xy=1 in the first quadrant.

Learn how the jacobian determinant governs the change of area under variable substitution, with direct and inverse mappings, including polar coordinates and the role of orientation.

Derive a single change-of-variables formula for double integrals using the jacobian determinant for direct and inverse substitutions, transforming the domain, integrand, and area element.

Compute an improper double integral over an unbounded region with x between 1 and 2, using y-first integration, yielding pi/2 times ln 2.

Learn how to compute triple integrals using Fubini's theorem, choosing among six orders of integration, applying iterated single integrals, and handling z-simple, x-simple, and y-simple domains.

Apply Fubini's theorem to evaluate the triple integral with constant limits, integrating in the order z, then y, then x, noting the integrand is independent of y.

Apply Fubini's theorem to compute the unit cube triple integral of y z^2 e^{-x y z}, choosing the order x then y then z, yielding 1/2 - 1/e.

Use Fubini on the z-simple domain defined by z≥0, x^2+y^2≤z^2, and x^2+y^2+z^2≤1, project to the disk of radius 1/√2, switch to polar coordinates, giving pi/8.

this lecture shows that areas and volumes can be computed in two ways and yield the same result: area by single or double integrals; volumes by double and triple integrals.

Master cylindrical coordinate change for a cone region, with theta from 0 to 2 pi and r from 0 to z, integrating z using the Jacobian.

Apply double and triple integrals to compute area between curves, volume between surfaces, and mass, center, and centroid of a domain, plus the area of a graph surface.

Compute the area of the surface z = 2x + 2y over the unit disk x^2 + y^2 ≤ 1, obtaining a constant integrand of 3 and area of 3π.

Explore field lines or streamlines as curves tangent to a vector field in 2D and 3D. Compute them via 2D determinants and 3D proportionalities, with an origin example using dy/dx.

Derive the streamline for a vector field with p = e^x and q = e^{-x}, reveal its independence from y, and obtain origin-passing streamline y = -1/2 e^{-2x} + 1/2.

Explore whether every vector field is a gradient by computing potentials. Use a paraboloid example and show that the field y minus x is not a gradient.

Show geometrically that a rotational vector field with circular field lines cannot be a gradient, since gradients point toward the fastest increase and cannot form a closed loop.

Determine whether the vector fields are conservative and compute their potentials; apply the necessary condition of equal mixed partials and verify results by differentiation.

Analyze a three-dimensional vector field to determine if it is conservative and, if so, find a potential; verify conditions with product and chain rules, and conclude it is not conservative.