Calculus III — Multivariable Calculus

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• 3D coordinates and distance ~10 min
• Vectors in space ~10 min
• Vector operations ~10 min
• The dot product ~12 min
• Projections and work ~12 min
• The cross product ~12 min
• Triple products and volume ~10 min
• Lines in space ~10 min
• Planes in space ~10 min
• Distances between points, lines, and planes ~12 min
• Cylinders and quadric surfaces ~14 min

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• Vector-valued functions ~10 min
• Limits and continuity of r(t) ~8 min
• Derivatives of vector functions ~10 min
• Differentiation rules for vector functions ~10 min
• Integrals of vector functions ~9 min
• Projectile and orbital motion ~12 min
• Arc length of a space curve ~9 min
• The unit tangent and arc-length parametrization ~10 min
• Curvature and the normal vector ~10 min
• TNB, torsion, and tangential/normal acceleration ~12 min

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• Functions of two and three variables ~10 min
• Domain and range ~12 min
• Graphs and surfaces ~12 min
• Level curves and contour maps ~12 min
• Level surfaces ~10 min
• Limits in two variables ~12 min
• Paths and nonexistent limits ~12 min
• Continuity of multivariable functions ~10 min
• A first look at partial rates ~10 min

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• Partial derivatives ~12 min
• The geometric meaning of a partial derivative ~10 min
• Higher-order partials and Clairaut's theorem ~12 min
• Differentiability and local linearity ~13 min
• Tangent planes ~11 min
• Linear approximation and differentials ~12 min
• The multivariable chain rule ~12 min
• Implicit differentiation in several variables ~11 min
• Taylor's theorem in two variables ~13 min
• When partials exist but the function isn't differentiable ~12 min

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• Directional derivatives ~12 min
• The gradient vector ~12 min
• The gradient as the direction of steepest ascent ~10 min
• The gradient is perpendicular to level curves ~10 min
• Gradients and level surfaces ~10 min
• Normal lines and tangent planes via the gradient ~12 min
• Gradient descent ~12 min
• The gradient in three variables ~10 min

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• Critical points ~10 min
• The second derivative test ~12 min
• Local versus absolute extrema ~9 min
• Extrema on closed, bounded regions ~13 min
• Applied optimization ~12 min
• Constrained optimization ~10 min
• Lagrange multipliers ~13 min
• Lagrange multipliers with two constraints ~12 min
• Least squares and applications ~13 min

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• Double integrals over rectangles ~10 min
• Iterated integrals and Fubini's theorem ~12 min
• Double integrals over general regions ~13 min
• Reversing the order of integration ~12 min
• Area and average value ~10 min
• Double integrals in polar coordinates ~13 min
• Surface area ~12 min
• Mass and density ~10 min
• Moments and center of mass ~12 min
• Moments of inertia ~12 min

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• Triple Integrals ~12 min
• General Solid Regions ~12 min
• The Six Orders of Integration ~12 min
• Cylindrical Coordinates ~12 min
• Spherical Coordinates ~13 min
• Choosing a coordinate system ~11 min
• The Jacobian ~13 min
• Change of variables in double integrals ~13 min
• Change of variables in triple integrals ~13 min

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• Vector fields ~10 min
• Gradient and conservative fields ~10 min
• Potential functions ~12 min
• Line integrals of scalar functions ~12 min
• Line integrals of vector fields and work ~12 min
• The Fundamental Theorem for line integrals ~10 min
• Independence of path ~10 min
• The 2D test for conservative fields ~10 min
• Circulation and flux in the plane ~12 min
• Green's theorem ~12 min

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• Parametric surfaces ~12 min
• Surfaces of revolution ~12 min
• Tangent planes to parametric surfaces ~12 min
• Surface area ~12 min
• Surface integrals of scalar functions ~12 min
• Orientation of a surface ~10 min
• Flux integrals ~12 min
• Applications of flux ~12 min

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• Divergence ~12 min
• Curl ~12 min
• The 3D conservative-field test ~12 min
• Green's theorem, two forms ~12 min
• Stokes' theorem ~14 min
• Using Stokes' theorem to change the surface ~12 min
• The divergence theorem ~14 min
• One theorem in four disguises ~12 min
• Applications in physics and electromagnetism ~14 min