Ace Advanced Calculus in 10.5 Hours (The Complete Course)

Learn the element of area for two-variable functions in Cartesian and polar coordinates, with dx dy as the Cartesian and r dr d theta as the polar area elements.

Compute area elements in generalized coordinates using the Jacobian determinant, via cross products of partials, with Cartesian and polar examples showing dA = |J| du dv and J = r.

Transform double integrals by changing variables with the Jacobian, translating x and y to u and v, and adjusting bounds; apply to Cartesian separable integrals and polar coordinates.

Use coordinate transformation in double integrals, with u=xy and v=x^2−y^2, to convert regions to rectangles, solve moment of inertia, and evaluate an exponential integrand.

Explore geometric and physical applications of double integrals, area, volume by rotation about axes, and centroid. Examin[e] mass with nonuniform density and moment of inertia.

Explore the gamma function from its integral definition, establish convergence for n>0, derive gamma(n+1)=n gamma(n) via integration by parts, and relate to n! for positive integers.

Derive gamma of one half from the gamma function definition using a variable substitution, yielding gamma(1/2) = sqrt(pi), and apply gamma(n+1) = n gamma(n) for higher halves.

Derives an equation to evaluate integrals of sin and cos using the gamma function, converting to a polar double integral and relating to gamma values.

Apply gamma function identities to evaluate cos and sin power integrals, illustrate with three examples, and use parity and quadrant rules to show the zero integral from 0 to pi.

Explore the Laplace transform, an integral transform with kernel e^{-st}, and derive its definition with examples like t^n and e^{at}. Understand convergence and gamma-based forms.

Explore triple integrals in cylindrical coordinates, using r, theta, and z to model a hurricane, compute total kinetic energy via density and velocity, and locate maximum velocity with Laplace transforms.

Transform triple integrals by rewriting x, y, z in terms of u, v, w, compute the jacobian via the scalar triple product, and apply cylindrical coordinates with jacobian r.

Demonstrates calculating the volume inside a sphere but outside a cylinder via triple integrals in cylindrical coordinates, with z bounds from the sphere and r from b to a.

Compute the volume cut from a sphere by the cylinder x^2+y^2=2x using cylindrical coordinates, with the cylinder centered at (1,0) and radius 1, and theta bounds -π/2 to π/2.

Compute the mass of the smaller part of a unit sphere cut by z = 1/2 using a cylindrical-coordinate triple integral with nonconstant density, switching integration order to obtain pi/4.

Study spherical coordinates using rho, phi, and theta. Transform from Cartesian coordinates and use the Jacobian rho^2 sin phi for triple integrals.

Compute the center of mass of a constant-density ice cream cone using triple integrals in spherical coordinates.

Calculate the planet's atmospheric mass with a triple integral in spherical coordinates using delta0 e^{-k(rho - a)}. The mass equals 4 pi delta0 (2/k^3 + 2a/k^2 + a^2/k).

Normalize a hydrogen atom wave function by enforcing its squared magnitude integrates to one with triple integrals in spherical coordinates. Derive k = sqrt(c^3 / pi).

Compute the moment of inertia and radius of gyration of a homogeneous solid sphere via triple integrals in spherical coordinates, deriving I = 8πδ a^5/15 and k^2 = (2/5)a^2.

Revisit triple integrals in spherical coordinates to set up the mass of a sphere portion cut by z=1/2, using rho and phi bounds with delta=1/rho, yielding pi/4.

Revisit lecture 22 using spherical coordinates to compute the same volume inside a sphere of radius a and outside a cylinder of radius b. Compute the bounds for rho, phi, and theta, evaluate the triple integral, and obtain the volume equals 4 pi over 3 times (a^2 - b^2)^(3/2).

Define surface area using a two-variable vector function r(u,v) and the area element dS = |∂r/∂u × ∂r/∂v| du dv, illustrated on spheres, cylinders, and cones with appropriate coordinates.

Derive the cartesian surface area using r(x,y) with z = f(x,y). Compute r_x cross r_y, yielding sqrt(f_x^2 + f_y^2 + 1), and integrate over the xy-projection.

Compute the surface area of a sphere in Cartesian coordinates by expressing z as a function of x and y, applying surface area formula, doubling for symmetry, obtaining 4πa^2.

Determine the surface area of the sphere cut by the cylinder x^2+y^2=y, using polar projection in the xy-plane and symmetry, yielding 2pi minus 4.

Relate height to radius in a flat cone using Cartesian coordinates, derive c(x,y) = (h/b) r, compute gradients, show the integrand is constant, and obtain SA = π b L.

Explore surface integrals in spherical coordinates using cross-product magnitude and a^2 sin phi, for a sphere of radius a with phi 0 to pi and theta 0 to 2 pi.

Examine surface integrals in spherical coordinates to determine the force on a point charge from a hollow sphere with uniform surface charge, comparing outside, inside, and on the surface cases.

Apply surface integrals in spherical coordinates to a solid sphere of density delta and compute force on point mass at b, yielding F = -G M_sphere / b^2 along k.

Define vector fields as quantities at every point in a domain, with examples like electric, gravitational, magnetic fields and velocity, and describe 2d streamlines via the unit tangent vector.

Explore flow examples to obtain streamlines by solving dy/dx = v2/v1: circular streamlines around a vortex and radial lines from a source, with flow direction set by omega or k.

Learn how scalar fields assign a quantity to every point and how gradients yield vector fields, revealing direction and magnitude of the greatest change and the level curves of constants.

Explore conservative vector fields and their potential functions by deriving a scalar psi from a given vector field, and identify which fields are conservative using gradient relationships.

Explore inverse-square law fields and determine when gravity and electric forces are conservative by deriving a scalar potential psi in spherical coordinates and applying Coulomb's law.

Study line integrals along smooth curves using Riemann sums and parametrization. Apply to curve length, mass, center of mass, moment of inertia, and surface area from rotation.

Explore line integrals of vector functions to compute work along curved paths, using unit tangent vectors and parametrization, and show conservative forces yield path-independent work via a potential function.

Compute the work from a variable attractive force proportional to distance to the origin along the parabola from (0,1) to (1,2). Include friction and sum the two works.

Verify a vector function is conservative by finding a potential function; for exact differential equations, rewrite the differential equation as the total differential of psi equal to zero.

Demonstrates the work-energy theorem: work equals the change in kinetic energy, with mv^2/2. Shows conservation for conservative forces via potential energy, while non-conservative forces like friction break it.

Compute flux and mass flow through surfaces by using velocity dot n and surface integrals, distinguishing closed and open surfaces and incorporating density for mass rate.

Compute the flux of velocity from a uniform point source at the origin. Show the flux equals c and velocity scales as 1/r^2 through a sphere of radius a.

Analyze a uniform dipole formed by a source and sink, derive its potential and velocity in spherical coordinates, and show the flux through a surrounding sphere is zero.

Derive the velocity field for a line source around a cylinder, yielding v equals m/r outward and a potential psi equals m ln r; flux equals the source rate c.

Explore flux and divergence by examining how a field's outward flow through a surface is measured, then localized at a point via the divergence ∇·F, with practical fluid examples.

Learn how circulation and the curl quantify rotation of a velocity field around a closed curve, using line integrals, unit tangent vectors, and the curl as a vector.

Define the fluid angular velocity as half the curl of its velocity field. Relate circulation around a closed curve and the limit as the enclosed area vanishes to derive omega.

Review the four vector operators: gradient, divergence, curl, and Laplacian, as the caption explains how scalars map to vector fields, vector fields to scalars, and the del operator.

Derive and apply vector identities using divergence, gradient, and curl for scalar times vector fields, including the product rule, curl of gradient, and the condition for conservative fields.

Examine vector identities and exact differential equations by verifying the curl of a vector field is zero and expressing the field as a gradient of a potential function.

Explore the divergence theorem, also called Gauss's theorem, which equates the volume integral of divergence to the flux across the enclosing surface, with a fluid velocity interpretation.

Derive the flux version of Green's theorem from the divergence theorem for a two-dimensional field in the x–y plane, linking the line integral around the curve to the area divergence.

Derive the scalar version of Green's theorem, using p and q, linking line integrals along curve C to a double integral over region R of ∂q/∂x − ∂p/∂y.

Explore the circulation version of Green's theorem for a planar vector field, linking the line integral to the curl's z-component, as a special case of Stokes' theorem in the plane.

Apply Green's theorem to multiply connected domains by partitioning the region into two simply connected subdomains, then combine their line and double integrals to cover the whole area.

Verify Green's theorem for a circle line integral using Green's function, showing the line integral equals 2 pi while the double integral equals 0 due to a singular origin.

Use Green's theorem to evaluate the line integral for any closed curve. It equals zero if the curve does not contain the origin, and 2 pi if it does.

Explore the line integral’s physical interpretation as flux through a unit-height cylinder, showing the flux equals k when the curve encloses the origin, otherwise zero.

Apply Green's theorem to convert the line integrals for work and circulation into a double integral over the region enclosed by curve C, equal to four times the area.

Apply Green's theorem to find the area enclosed by a closed curve; turn the double integral into a line integral using suitable p and q.

Two useful results from Green's theorem show that if p depends only on x with q zero, or p zero with q depending only on y, the line integrals vanish.

Compute flux and circulation for two velocity fields over an ellipse, using divergence and Stokes theorems; find zero flux for the vortex-like field and nonzero circulation.

Derive Gauss's law in the plane by comparing a unit line source with an inverse radial field; flux through a curve enclosing the origin is 2 pi k, otherwise zero.

Verify the divergence theorem for a region bounded by a radius two cylinder, a paraboloid z=r^2, and z=0, using cylindrical coordinates; show volume integral equals the surface flux 16 pi.

Apply the divergence theorem to a multiply connected domain, equating the volume integral of div f to the flux through S1 minus the flux through S2 with outward normals.

Apply Gauss's theorem to a radially outward field proportional to 1/ρ^2 and show zero flux for surfaces not enclosing the origin, or 4πc if they do.

Apply Gauss' theorem to inverse-square fields from a uniform point source and the electric field, linking flux to enclosed charge. Maxwell's equations relate these Gauss laws.

Derive and interpret Stokes' theorem for three-dimensional fields by linking the curl over a surface to the boundary circulation, generalizing from the xy plane to space.

Verify Stokes theorem by equating a surface integral of curl f over a paraboloid capped by z=2 to a clockwise line integral around its boundary, using polar coordinates and orientation.

Investigate two special cases of Stokes theorem: flux of curl on a closed surface via the divergence theorem, and the flux of curl over a planar region and its boundary.

Apply Stokes' theorem to a closed surface formed by a paraboloid and the plane z=2, showing the planar flux is -20 pi and the total flux is zero.

Shows that zero curl implies a conservative field, using Stokes theorem to prove path-independent line integrals and a gradient of a scalar potential.

Introduce the fundamental lemma, proving a continuous function with zero volume integral over every region must vanish, and derive key PDEs in fluid, thermal, and electromagnetism.

Derive the equation of continuity from conservation of mass, linking the rate of density change and fluid flux to sources and sinks within a closed volume.

Explore the equation of continuity and special cases: incompressible flow, irrotational flow with velocity as the gradient of a potential, and Poisson's and Laplace's equations for sources.

Derive the diffusion of heat from conservation of energy using Fourier diffusion equation, linking temperature psi to specific heat capacity, density, and thermal conductivity in thermophysics.

Derive the four Maxwell's equations from integral theorems and field relations in electromagnetic theory, including Gauss's laws for electric and magnetic fields, Faraday's law, and Ampere's law.