Part VI: Calculus III
Vectors, surfaces, gradients, multiple integrals, and the theorems of vector calculus.
- Module 33Vectors, lines, planes, and surfaces in space8 exercisesUse vector arithmetic and products, construct lines and planes, compute geometric distances and areas, and identify basic quadric surfaces from traces.
- Module 34Vector-valued motion, arc length, and curvature8 exercisesCompute spatial velocity and acceleration, parameterize by arc length in simple cases, and separate tangential acceleration from turning.
- Module 35Multivariable functions, level sets, and limits8 exercisesRead a function with several inputs, sketch a domain or level set, disprove limits by paths, and prove limits with bounds that work in every direction.
- Module 36Partial derivatives, differentiability, and the chain rule8 exercisesCompute partial and mixed derivatives, distinguish them from total differentiability, and track all dependency paths in a multivariable chain rule.
- Module 37Gradients, directional derivatives, and tangent planes8 exercisesNormalize a direction, compute a directional rate, interpret the gradient, and build a tangent plane or local approximation from the correct point and derivatives.
- Module 38Multivariable extrema and constrained optimization8 exercisesClassify regular critical points, inspect boundaries on compact regions, and use Lagrange multipliers without losing singular or boundary candidates.
- Mixed review R3: approximation, geometry, and local change8 exercisesUse this review after Module 38. The questions bring series and curve geometry back into multivariable work. Do not treat an unfamiliar representation as evidence that the underlying mathematics is new. The magnitudes…
- Module 39Double integrals, changing order, and polar area elements8 exercisesDescribe a region with correct variable bounds, evaluate iterated integrals, reverse order when useful, and include the polar area factor.
- Module 40Triple integrals and cylindrical and spherical coordinates8 exercisesDescribe a solid with nested bounds, choose cylindrical or spherical coordinates, and compute volume, mass, or moments with the correct volume element.
- Module 41Jacobians and general changes of variables8 exercisesCompute a Jacobian, transform both a region and an integrand, use absolute volume scaling, and recognize when a coordinate map double-counts a region.
- Module 42Vector fields, scalar line integrals, work, and circulation8 exercisesDistinguish scalar fields from vector fields, choose between dd s and ddvv r, compute work or wire mass, and track orientation.
- Module 43Conservative fields and Green’s theorem8 exercisesFind a potential when it exists, use endpoint evaluation for conservative work, and apply Green’s circulation or flux form with domain and orientation conditions.
- Module 44Surface area, scalar surface integrals, and flux8 exercisesParameterize a surface patch, compute its area element, choose an oriented vector area element, and distinguish an open surface from a closed boundary.
- Module 45Curl, divergence, and Stokes’ theorem8 exercisesCompute curl and divergence as different objects, connect circulation to curl flux, and apply Stokes with a compatible orientation and a nonsingular spanning surface.
- Module 46The Divergence theorem and a unified view of calculus8 exercisesConvert closed-surface flux to a volume integral, close an open surface with explicit corrections, handle cavities and singularities, and select the appropriate boundary theorem.
- Cumulative assessment: Calculus III12 exercisesUse this after Module 46. Begin each geometric integral by naming the object: curve, surface, or solid. State the domain, orientation, and theorem conditions before using a shortcut. Work all questions before viewing the…
- chapter
- workshop: connects two subjects
- checkpoint, review or assessment