Multivariable Calculus
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
19 lessons
- Three-Dimensional Coordinates, Distance and SpheresLocate points in space with three coordinates, find distances and midpoints, write the equation of a sphere, and read simple equations in x, y and z as surfaces.
- The Cross ProductMultiply two vectors in space to get a third, perpendicular to both, whose length is the area of the parallelogram they span. Includes the triple product and torque.
- Lines and Planes in SpaceDescribe a line in space with a point and a direction, a plane with a point and a normal vector, and find intersections, angles and distances between them.
- Functions of Several Variables and Contour MapsFunctions with two or three inputs: their domains, their graphs as surfaces, and level curves, which draw a surface on flat paper the way a topographic map does.
- Partial DerivativesDifferentiate a function of several variables in one variable at a time, holding the rest fixed. Read partials as slopes of slices, and compute second partials.
- Tangent Planes and Linear ApproximationBuild the plane that touches a surface z = f(x, y) at a point from the two partial derivatives, use it to approximate f nearby, and estimate errors with differentials.
- The Multivariable Chain RuleDifferentiate a function of several variables along a path, or through a change of variables, by adding one term per route, and differentiate implicit curves.
- Directional Derivatives and the GradientFind the rate of change of f in any direction as a dot product with the gradient, see why the gradient points uphill, and use it as a normal to level curves and surfaces.
- Maxima, Minima and Saddle PointsFind the critical points of f(x, y), classify them with the second derivative test, and find absolute extremes on a closed region by checking its boundary too.
- Lagrange MultipliersOptimize f(x, y) along a constraint curve g(x, y) = k by solving ∇f = λ∇g, and see why the best point is where a level curve of f touches the constraint.
- Double Integrals over RectanglesFind the volume under a surface z = f(x, y) with Riemann sums of boxes, then compute double integrals exactly as iterated integrals using Fubini's theorem.
- Double Integrals over General RegionsIntegrate over regions bounded by curves: set up type I and type II limits, compute areas and volumes, and reverse the order of integration to unlock hard integrals.
- Double Integrals in Polar CoordinatesIntegrate over disks, rings and wedges with dA = r dr dθ, see where the extra factor r comes from, and use it to evaluate the Gaussian integral.
- Triple IntegralsIntegrate a function over a solid region in space: set up limits from the inside out, find volumes and masses, and locate the center of mass of a solid.
- Triple Integrals in Cylindrical and Spherical CoordinatesDescribe cylinders, cones and balls with cylindrical and spherical coordinates, and integrate over them using dV = r dz dr dθ and dV = ρ² sin φ dρ dφ dθ.
- Vector FieldsA vector field attaches an arrow to every point: wind, flowing water, gravity. Sketch fields, compute gradient fields and follow flow lines.
- Line IntegralsIntegrate along a curve: find the mass of a wire with ∫f ds, and the work a force field does along a path with ∫F · dr, using a parametrization of the curve.
- The Fundamental Theorem for Line IntegralsFor a gradient field, a line integral is the change in the potential between the endpoints. Test whether a field is conservative and find its potential.
- Green's TheoremGreen's theorem turns a line integral around a closed curve into a double integral over the region inside. Use it to compute circulation and area.