Multivariable Calculus

Explore functions of two or more variables, where x and y are independent and z is dependent, determine domains, and sketch level curves and level surfaces in three dimensions.

Explore limits and continuity in functions of two variables. Determine whether the limit as (x,y) approaches (0,0) equals L by examining multiple paths and using informal definitions and examples.

Learn partial derivatives of functions of two variables, defining F_x, F_y, Z_x, Z_y, and their geometric interpretation as slopes of tangent lines; practice second and mixed derivatives and Clairaut's theorem.

The lecture generalizes differentiability to functions of two variables, shows partial derivatives alone don’t guarantee differentiability or continuity, and introduces the total differential and tangent planes.

Explore the chain rule from single-variable to multivariable functions, using partial derivatives, two-variable and three-variable forms, and implicit differentiation with practice examples.

Compute directional derivatives by dotting the gradient with a unit vector, showing the maximum rate equals the gradient's length. The gradient is normal to level surfaces.

Explore how to locate relative and absolute maxima and minima for functions of two variables on closed regions, using critical points, partial derivatives, the Hessian, and boundary analysis.

Introduces double integrals to measure volume under a surface over region r in the x-y plane, using iterated integration and interchanging the order of integration.

Explore double integrals over nonrectangular regions by sketching boundaries, identifying type one and type two regions, and changing the order of integration to evaluate difficult limits, compute volumes and areas.

Learn to set up and evaluate integrals over polar regions by converting to r–theta iterated integrals, including origin-touching cases, with examples of a circle vs cardioid and a rose curve.

Explore triple integrals as the three-dimensional extension of single and double integrals, integrating f(x, y, z) over a solid G using iterated or double-triple integrals to compute volume.

Master cylindrical and spherical coordinates, convert between rectangular, cylindrical, and spherical systems, and visualize points as intersections with cylinders, cones, and spheres.

Explore cylindrical and spherical triple integrals, set up with appropriate limits, including the Jacobian factors R and rho^2 sin phi, through sphere volume and cone-sphere intersection examples.

Explore vector fields in two and three dimensions, visualize field vectors, compute divergence and curl using del notation, and connect gradient fields to conservative potentials with examples.

Explore line integrals of functions f(x,y) along curves, using parameterizations x(t), y(t) and ds, dx, dy, to turn into definite integrals with examples like a quarter circle.

Learn how line integrals in conservative vector fields are independent of the path, using potential functions, the fundamental theorem of work integrals, and the curl test in simply connected regions.

Apply Green's theorem to convert line integrals to double integrals for simply connected and multiply connected regions; master area formulas and ellipse area calculations through worked examples.

Explore parametric surfaces in 3d space with two parameters u and v, using vector valued functions r(u,v) to form tangent planes via partial derivatives and cross products.

Learn to compute surface integrals by parameterizing surfaces in three dimensions, converting to double integrals via the cross product magnitude, and applying to surface area and general functions.

Study surface orientation and flux: two possible orientations, gradient-based and cross-product methods, and practical examples with sphere and ellipsoid flux calculations.

Explore Stokes theorem and its relation to Green's theorem, linking line integrals around closed curves to surface integrals of curl over oriented surfaces, with practical examples.

Apply the divergence theorem to convert surface flux into triple integrals over simple regions, illustrating with the unit sphere, unit cube, and paraboloid.