Calculus 3 (multivariable calculus), part 2 of 2

Explore the repetition of Riemann integrals, definition and notation of definite integrals, and how single integrals relate to multiple, line, and surface integrals through partitions and sums.

Explore how Riemann sums approach the integral for functions on an interval, and why the rationals indicator is not integrable, with upper sums at 1 and lower sums at 0.

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 core methods for computing Riemann integrals, guided by the fundamental theorem of calculus, including integration by parts, substitution, reverse derivatives, and partial fractions, with emphasis on primitive functions.

Explore how curves are described mathematically with vector-valued and parametric representations, linking plane and space curves to time and velocity as tangent, and connect this view to Riemann integrals.

Explore arc length for plane and space curves using a Riemann construction and vector-valued functions. Relate speed, distance, and dt to derive arc length as the integral of velocity magnitude.

Explore sets in the plane, including axis-parallel rectangles (apr), and learn how domains guide double integrals. Visualize x- and y-boundaries and vertical or horizontal cross-sections.

Learn to use double integrals to compute the signed volume over a compact, closed, and bounded domain in the plane, with dx dy as the area element and applications.

Explore three ways to describe an axis-parallel rectangle for double integrals: vertex coordinates, Cartesian product of intervals, and inequalities, and learn how the domain guides computation.

Define double integrals on axis-parallel rectangles. Build Riemann sums with partitions in x and y, yielding the volume between the surface and the x y plane.

Embed domain into axis-parallel rectangle to define double integrals on compact domains. Define f hat as f on D and zero outside, enabling Riemann sums to compute the double integral.

Explore multiple integrals from single and double to triple integrals, using Riemann sums and axis-parallel blocks to define volume integrals for functions of x, y, z.

Explore the properties of double integrals, including linearity, monotonicity, and the triangle inequality, and explain how to partition domains and apply these rules to compute signed volumes between surfaces.

Estimate double integrals by inspection using geometrical reasoning, treating the double integral as signed volume between the graph surface and the xy-plane over rectangles, disks, and triangles.

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.

Explore integration by inspection for two-variable odd functions, using domain symmetry about the x or y axis to make double integrals vanish, via positive and negative volume cancellation.

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.

this lecture shows integration by inspection for the domain |x|+|y| ≤ pi, proving the double integral of sin(x+y) over d is zero by geometric symmetry and the sine addition formula.

Apply Fubini's theorem to axis-parallel rectangles by evaluating the double integral of x^3 y^2 over [0,1]×[0,2] using both orders of integration, illustrating that both yield 2/3.

Learn a practical rule: Fubini's theorem for axis-parallel rectangles, where a continuous function factors as g(x) h(y), and the double integral equals the product of the single integrals.

Apply Fubini's theorem to compute the double integral over x in [0,1], y in [1,2], of e^(xy)(1+xy). Use integration by parts on the inner x-integral to obtain e^2 - e.

Demonstrates a final example where the order matters in Fubini's theorem for a double integral on an axis-parallel rectangle, showing that integrating with respect to y first simplifies the computation.

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

Learn how to apply Fubini's theorem for x- and y-simple domains by performing double integration via iterated integrals, with axis-parallel rectangles and variable limits.

Using Fubini's theorem with a y-simple domain, we integrate f(x,y)=2xy over 0≤x≤2 and -x≤y≤2x to obtain 12.

Use Fubini's theorem to evaluate the double integral of x y over the first-quadrant domain between y = x^2 and y = x. Both y-simple and x-simple methods yield 1/24.

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.

Apply Fubini's theorem to a double integral over an x-simple vs a y-simple domain; show that choosing the easier order shortens the solution and yields one fourth of ln 2.

We describe the domain as an x-simple region between sqrt(1+y^2) and sqrt(9−y^2) for y in [−2,2], then apply Fubini's theorem to compute the double integral of x^3 y^2, yielding 128/3.

Apply Fubini's theorem to rewrite the domain as y-simple, compute the inner integral with respect to y, and finish by evaluating the outer integral to obtain (e^9 - 1)/6.

Apply Fubini's theorem to the general double integral of e^{y^3} on the domain y in [0,1], x in [0, y^2]. Switch the order and obtain the result 1/3 (e−1).

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.

Explore why change of variables in double integrals simplifies computations, compare calc two and calc three approaches, and preview direct and inverse substitutions and the jacobian with polar coordinates.

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.

Transform disk domains to polar coordinates using inverse substitution, apply the Jacobian, and compute volumes under z = 1 - x^2 - y^2 in multivariable calculus.

Use direct substitution to transform the double integral of e^{x+y} over a square domain by u=x+y and v=x−y, compute the Jacobian, and compare the inverse function theorem with explicit inversion.

This lecture solves a change-of-variables problem for a double integral over the upper half-disk of radius one, using polar coordinates and a product-based separation.

Convert the integral to polar coordinates on the region between radii 1 and sqrt(2); theta integral gives 2π, then use u = 1 + r^2 to obtain π[log(27/4) − 1].

Compute the volume between the paraboloids z = x^2 + y^2 and z = 4/3 − (x^2+y^2)/3 by integrating over the unit disk, yielding 2π/3.

Apply direct substitution with u = x^2 - y^2 and v = x y to transform the domain into a rectangle. Compute the double integral via the Jacobian and Fubini.

Change of variables u = x y and v = y / x transforms domain to axis-parallel rectangle; compute area with a double integral to get 3/2 log 2.

Wraps up double integrals by revisiting Fubini's theorem and iterated integrals over axis-parallel rectangles, and emphasizes change of variables, including polar coordinates, for roundish domains and level curves.

Explore improper integrals, including unbounded domains and unbounded integrands, and learn convergence criteria p>1 and q<1 for p- and q-integrals, plus splitting divergent improper integrals.

Explore improper double integrals in multivariable calculus, handling unbounded and bounded domains with exhaustive sequences of sets, convergence criteria, and splitting into f plus and f minus.

Show how calc 3 helps calc 2 by computing the improper integral of e^{-x^2} over R via a double integral, using Fubini and polar coordinates to reach sqrt(pi).

Compute the improper double integral over the first quadrant by expanding over growing squares and splitting into two one-variable integrals, yielding convergence with value pi^2/4.

Investigate an improper double integral over a bounded triangle where 0 < x < y < 1 and the integrand 1/(y-x) diverges near y = x, proving non-convergence.

Solve an improper double integral over the unit disk of log(x^2+y^2) using polar coordinates; show convergence to -pi and interpret as negative volume under the xy-plane.

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.

Apply the mean value theorem for double integrals to continuous functions on compact, path-connected domains, proving a point where the integral equals f(x0,y0) times the area.

Apply the mean value theorem for double integrals to a triangular half-square domain, compute the double integral of x^2+y^2, and divide by the domain area to find the mean value.

Apply the mean value theorem for double integrals to f(x,y)=1/x over the domain x in [0,1], y between x^2 and sqrt(x), using Fubini and improper integral and area for normalization.

Explore the notation, definition, and properties of triple integrals, including Riemann sums over axis-parallel boxes, domain embedding, and linearity, monotonicity, triangle inequality, with the volume element dx dy dz.

Explore integration by inspection in multivariable calculus, using geometrical reasoning to estimate triple integrals of constants as volumes. Use symmetry and oddness with respect to x, y, and z to cancel terms, as shown for the upper half of a ball (radius two), yielding 16 pi.

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.

Apply Fubini's theorem to compute the triple integral of x over a tetrahedron bounded by x=1, y=1, z=1, and x+y+z=2. Evaluate the integral to obtain 1/8.

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.

Show three ways to find a tetrahedron’s volume with Fubini’s theorem: base area times height divided by three, double integral of one minus x minus y, and a triple integral.

Control domain transformation from x,y,z to u,v,w using the Jacobian determinant in triple integrals. Use inverse and direct substitutions to convert between coordinate systems such as cylindrical and spherical.

Revisit a triple integral over an ice cream cone, applying Fubini with cylindrical coordinates; switch to spherical coordinates to obtain a box, enabling a product of single-variable integrals with Jacobian.

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 change of variables to compute the volume between a paraboloid and a cone using cylindrical coordinates, determine the intersection radius, and justify cylindrical over spherical coordinates.

Compute a triple integral over the ball of radius two using spherical coordinates. Use the Jacobian and Fubini to obtain 4pi(2 - arctan 2).

Solve problem five via direct substitution, transforming the parallelepiped to an axis-aligned uvw box where the integrand becomes v, and compute the jacobian to obtain the value 16.

Explore the change of variables for triple integrals, transforming complex domains into axis-parallel boxes, and apply jacobian determinants with direct and inverse substitutions using cylindrical or spherical coordinates.

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 mass with density rho by applying double integrals to planar domains and triple integrals to solids, interpreting density as locally constant so mass equals density times area or volume.

Apply triple and double integrals to find the mass center and centroid of solids and plane regions, using density, total mass, and symmetry to compute coordinates.

Derive the area formula for a graph surface z=f(x,y) over a compact domain D using a double integral of sqrt(1+f_x^2+f_y^2), and illustrate with the upper half of a sphere.

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.

Compute the surface area of the paraboloid z = 4 − x^2 − y^2 above the xy-plane using polar coordinates, obtaining area = (π/6)(17√17 − 1).

Compute the surface area of the conic surface 3 z^2 = x^2 + y^2 for z in [0,2] by integrating over x^2 + y^2 ≤ 12 to obtain 8√3 π.

Compute the area of the parabolic cylinder z = y^2 over the triangle 0 ≤ x ≤ y, 0 ≤ y ≤ 1 using Fubini; result is 1/12(5√5-1).

Explore the four function types in multivariable calculus: real-valued, vector-valued, multivariable scalar fields, and vector fields, and visualize them with parametric representations and polar coordinates.

Explore vector fields with intuitive pictures and examples, including gradient fields, wind velocity, electric and magnetic fields, and field lines, linking visuals to calculus concepts.

Treat vector fields as real objects with domains and component functions p and q (and r in 3D). Plot them; domain is the intersection of component domains; note C1/C2 smoothness.

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.

Solve streamlines for the planar vector field f(x,y)=(2x,2y) using determinant condition and separable ODE; find axis halves and all straight lines through the origin as streamlines.

Solve problem 2 by deriving a separable ode from the determinant, showing streamlines satisfy x^2+y^2 = d with d >= 0, i.e., circles centered at the origin.

Solve problem 3 on streamlines by turning the determinant form into a separable equation, yielding xy = const as hyperbola branches and noting the axes and origin as streamlines.

Swap coordinates, form a determinant equal to zero to get the separable differential equation x dx = y dy, leading to x^2 − y^2 = d, diagonals, hyperbola streamlines.

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.

Enforce parallelism of velocity and field to solve 3d streamline problem; derive separable relations giving x equals C1 e^{-1/y} and z equals C2 e^{-1/y}, using y as parameter.

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.

Explore how conservative vector fields are gradients of scalar potentials, and how equipotential lines arise as level curves of the potential, orthogonal to the field.

Explore Schwarz's theorem on the equality of mixed partials for C2 functions, and see how continuous second partials yield a symmetric Hessian, clarifying the Jacobian–Hessian relationship.

Compare Hessian and Jacobian matrices in multivariable calculus, highlighting second-order derivatives and symmetry for C2 functions, and explain Jacobian as gradients of component functions for vector fields.

Check the necessary (not sufficient) condition for a conservative field by equating mixed partials: in two dimensions, ∂p/∂y = ∂q/∂x; in three dimensions, corresponding equalities to help find a potential.

Explore how to find a potential function for a two-dimensional electrostatic field, showing its conservative nature and deriving the gradient condition using the chain rule and change of variables.

Explore the gravitational field as a conservative field with a potential function, derive the inverse-square force, and verify the gradient of phi equals the gravitational field.

Assess whether vector fields are conservative using the mixed partials condition, compute a potential when possible, and verify by differentiating the gradient.

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

determine whether the three-dimensional vector field is conservative and, if so, obtain its potential using the p, q, r conditions.

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.