https://teddyrookbook.com/math/calculus-3/Recent content in Calculus 3: Multivariable Calculus on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/calculus-3/space-coordinates/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/space-coordinates/Locate points with (x, y, z) and find distances in space. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/vectors-in-space/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/vectors-in-space/Add, scale and find the length of 3D vectors. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/dot-product-projections/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/dot-product-projections/u · v = |u||v| cos θ, and the projection of one vector on another. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/cross-product/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/cross-product/u × v is perpendicular to both, with length equal to the parallelogram's area. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/lines-in-space/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/lines-in-space/r(t) = r₀ + tv. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/planes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/planes/A normal vector and a point give a(x − x₀) + b(y − y₀) + c(z − z₀) = 0. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/distance-to-plane/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/distance-to-plane/Project onto a normal to find the shortest distance. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/quadric-surfaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/quadric-surfaces/Identify surfaces by their traces in the coordinate planes. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/cylindrical-spherical/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/cylindrical-spherical/Describe points by (r, θ, z) or (ρ, θ, φ). Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/vector-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/vector-functions/r(t) traces a curve as t changes. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/vector-function-calculus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/vector-function-calculus/Differentiate and integrate component by component. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/space-arc-length/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/space-arc-length/L = ∫|r′(t)| dt. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/curvature/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/curvature/Curvature measures how fast the direction turns. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/motion-in-space/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/motion-in-space/Split acceleration into tangential and normal parts. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/multivariable-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/multivariable-functions/Read surfaces and level curves of z = f(x, y). Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/level-surfaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/level-surfaces/f(x, y, z) = k describes a surface. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/multivariable-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/multivariable-limits/A limit must agree along every path. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/partial-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/partial-derivatives/Differentiate in one variable while holding the others fixed. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/higher-partials/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/higher-partials/Mixed partials fxy and fyx agree for smooth functions. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/tangent-planes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/tangent-planes/z ≈ f(a, b) + fx(x − a) + fy(y − b). Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/differentials-error/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/differentials-error/Estimate how small errors in inputs affect the output. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/multivariable-chain-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/multivariable-chain-rule/Add up the effects through every path of dependence. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/implicit-partials/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/implicit-partials/dy/dx = −Fx/Fy. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/directional-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/directional-derivatives/The rate of change in the direction of a unit vector u. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/gradient/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/gradient/∇f points uphill fastest, with length equal to the steepest rate. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/gradient-normals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/gradient-normals/∇f is perpendicular to level curves and level surfaces. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/critical-points/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/critical-points/Use D = fxx·fyy − (fxy)² to classify critical points. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/absolute-extrema/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/absolute-extrema/Check critical points inside and the whole boundary. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/lagrange-multipliers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/lagrange-multipliers/At a constrained extreme, ∇f is parallel to ∇g. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/lagrange-two-constraints/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/lagrange-two-constraints/∇f = λ∇g + μ∇h. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/gradient-descent/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/gradient-descent/Step against the gradient, x ← x − η∇f, to walk downhill toward a low point. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/minimizing-loss/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/minimizing-loss/Choose a line's slope and intercept to make the mean squared error as small as possible. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/backpropagation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/backpropagation/Get every weight's derivative of a network's loss by working backward, one layer at a time. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/double-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/double-integrals/Volume under a surface as a limit of sums. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/iterated-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/iterated-integrals/Integrate one variable at a time. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/general-regions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/general-regions/Let the inner limits depend on the outer variable. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/reverse-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/reverse-order/Redraw the region and describe it the other way. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/polar-double-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/polar-double-integrals/dA = r dr dθ. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/double-integral-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/double-integral-applications/Integrate density over a region. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/surface-area-double/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/surface-area-double/A = ∬√(1 + fx² + fy²) dA. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/triple-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/triple-integrals/Integrate over solid regions. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/triple-cylindrical/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/triple-cylindrical/dV = r dz dr dθ. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/triple-spherical/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/triple-spherical/dV = ρ² sin φ dρ dφ dθ. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/triple-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/triple-applications/Find mass, centers of mass and moments of inertia. Calculus 3: Multivariable Calculus · Math Ladderhttps://teddyrookbook.com/math/calculus-3/change-of-variables/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-3/change-of-variables/Transform a region and scale by |∂(x, y)/∂(u, v)|. Calculus 3: Multivariable Calculus · Math Ladder