Partial derivatives, multiple integrals, and vector fields
← Back to MathematicsUnderstand and visualize functions of multiple variables in 3D space.
Master partial differentiation and understand geometric interpretations.
Apply chain rule to composite functions and solve implicit problems.
Explore directional derivatives, gradients, and optimization.
Find extrema using critical points and Lagrange multipliers.
Integrate functions over regions in the xy-plane.
Use polar coordinates for double integrals over circular regions.
Extend integration to three dimensions and various coordinate systems.
Understand vector fields and their fundamental properties.
Master line integrals, surface integrals, and fundamental theorems.
Learn to work with and visualize functions that depend on multiple input variables.
Define and understand functions of two, three, and more variables with their domains and ranges.
Master the 3D rectangular coordinate system and learn to plot points and surfaces in space.
Create and interpret contour maps and level curves for functions of two variables.
Visualize and analyze level surfaces for functions of three variables.
Understand limits and continuity for multivariable functions using precise definitions.
Learn various methods for sketching and understanding graphs of multivariable functions.
Master the concept of partial differentiation and its geometric interpretations.
Understand the definition and notation of partial derivatives for multivariable functions.
Visualize partial derivatives as slopes of tangent lines to curves on surfaces.
Compute second and higher-order partial derivatives, including mixed partials.
Understand when mixed partial derivatives are equal and apply Clairaut's theorem.
Find equations of tangent planes to surfaces using partial derivatives.
Use tangent planes and total differentials for linear approximation and error estimation.
Apply the chain rule to composite multivariable functions and solve implicit problems.
Master the chain rule for functions of several variables with multiple intermediate variables.
Use tree diagrams to organize and apply the chain rule systematically.
Find partial derivatives of implicitly defined functions using implicit differentiation.
Solve related rates problems in multivariable contexts using the chain rule.
Work with parametrically defined surfaces and compute their derivatives.
Apply chain rule concepts to change of variables in differentiation problems.
Understand how functions change in any direction and find paths of steepest ascent.
Compute the rate of change of functions in any specified direction using unit vectors.
Understand the gradient as a vector of partial derivatives and its geometric meaning.
Explore key properties of gradients including their relationship to directional derivatives.
Understand how gradients are perpendicular to level curves and level surfaces.
Find directions of steepest ascent and descent using gradient vectors.
Apply directional derivatives and gradients to optimization and physics problems.
Find maximum and minimum values of multivariable functions with and without constraints.
Find critical points by setting partial derivatives equal to zero.
Use the discriminant and Hessian matrix to classify critical points.
Find absolute maximum and minimum values on closed and bounded regions.
Solve constrained optimization problems using the method of Lagrange multipliers.
Handle optimization problems with multiple constraints using multiple Lagrange multipliers.
Apply optimization techniques to problems in economics, engineering, and physics.
Integrate functions of two variables over regions in the plane.
Evaluate double integrals over rectangular regions using iterated integrals.
Understand when and how to change the order of integration in double integrals.
Set up and evaluate double integrals over more complex, non-rectangular regions.
Choose the best order of integration and change orders when beneficial.
Use double integrals to find areas, volumes, average values, and centers of mass.
Apply double integrals to joint probability density functions and expected values.
Use polar coordinates to simplify double integrals over circular regions.
Convert between Cartesian and polar coordinates for double integrals.
Understand and apply the Jacobian factor r in polar coordinate transformations.
Set up and evaluate double integrals in polar coordinates with proper limits.
Apply polar coordinates to problems involving circles, annuli, and sectors.
Calculate volumes of solids with circular symmetry using polar double integrals.
Find mass and centroid of laminas with circular boundaries using polar coordinates.
Extend integration to three dimensions using rectangular, cylindrical, and spherical coordinates.
Set up and evaluate triple integrals in standard xyz-coordinate system.
Use triple integrals to find volumes of 3D regions and mass of solid objects.
Transform to cylindrical coordinates (r,θ,z) for problems with circular symmetry.
Use spherical coordinates (ρ,θ,φ) for problems involving spheres and cones.
Choose the most appropriate coordinate system based on the geometry of the region.
Calculate moments of inertia for 3D objects using triple integrals.
Understand vector fields and their fundamental properties including divergence and curl.
Visualize and understand vector fields in 2D and 3D space with direction and magnitude.
Recognize and work with gradient fields and their relationship to scalar functions.
Identify conservative vector fields and find their potential functions.
Compute divergence of vector fields and understand its physical meaning as source/sink strength.
Calculate curl of vector fields and interpret it as measure of rotation or circulation.
Apply vector field concepts to fluid flow, electromagnetism, and gravitational fields.
Master line integrals, surface integrals, and the fundamental theorems of vector calculus.
Evaluate line integrals of scalar functions along curves in 2D and 3D space.
Compute work done by vector fields along paths and understand path independence.
Apply Green's theorem to relate line integrals around closed curves to double integrals.
Integrate scalar functions over surfaces in 3D space using parameterizations.
Calculate flux of vector fields through surfaces and understand orientation.
Connect line integrals around closed curves to surface integrals using Stokes' theorem.
Relate surface integrals to triple integrals using the divergence theorem (Gauss's theorem).
Apply these theorems to problems in physics, engineering, and fluid dynamics.