Courses / Mathematics / Multivariable calculus: geometry, change and accumulation Mathematics · unit version 1 Multivariable calculus: geometry, change and accumulation Twenty substantial lessons in three-dimensional geometry, multivariable differentiation, integration and vector-calculus theorems. Exact exercises accompany reviewed setup and proof tasks. This authored practice sequence does not by itself establish assessed Calculus III preparation; cumulative assessment, source audit and learner evidence remain distinct.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
For vectors a=(1,2,−1) and b=(2,0,3), find a·b and ||a||. Vector lengths, projections and normal directions appear throughout spatial calculus.
Compare reasoning and find a refresher The dot product is 2+0−3=−1. The length of a is √(1+4+1)=√6. A negative dot product indicates an obtuse angle between nonzero vectors.
Review dot products and projections
Differentiate (1+3t²)^4 and explain the inner rate. Multivariable chain rules extend the same composition mechanism to several changing inputs.
Compare reasoning and find a refresher The derivative is 4(1+3t²)^3·6t=24t(1+3t²)^3. The factor 6t is the rate of the inner function.
Review the chain rule
Evaluate ∫₀² 3x² dx. What changes when you reverse the bounds? Multiple and line integrals depend on reliable single-variable integration and orientation.
Compare reasoning and find a refresher The integral is [x³]₀²=8. Reversing the bounds gives −8; it changes the directed accumulation, not the integrand.
Review definite integration
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Multivariable calculus: geometry and theorem choice
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Spatial transport dossier: one vessel, three descriptions
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Use three-dimensional vectors to construct lines, normals, planes and volumes.
Use three-dimensional vectors to construct lines, normals, planes and volumes. Construct lines and planes from points and direction or normal vectors. Check perpendicularity, intersection and volume using the correct vector product. 6 practice tasks.
Lesson details and objectives Separate velocity from speed and compute arc length and curvature of regular curves.
Separate velocity from speed and compute arc length and curvature of regular curves. Compute speed and distance traveled independently of displacement. Construct tangent directions and distinguish tangential from normal acceleration. 6 practice tasks.
Lesson details and objectives Read surfaces through slices and distinguish necessary path tests from a genuine limit argument.
Read surfaces through slices and distinguish necessary path tests from a genuine limit argument. Disprove a limit using incompatible approach paths. Prove a limit with an all-path bound rather than a few sampled paths. 6 practice tasks.
Lesson details and objectives Use partials to build tangent planes while checking the stronger differentiability requirement.
Use partials to build tangent planes while checking the stronger differentiability requirement. Construct a tangent plane and linear approximation at a differentiable point. Distinguish existence of partial derivatives from continuity and differentiability. 6 practice tasks.
Lesson details and objectives Compose derivative maps and distinguish a unit-direction rate from a parameter-dependent rate.
Compose derivative maps and distinguish a unit-direction rate from a parameter-dependent rate. Normalize a direction before taking its dot product with a gradient. Compare a directional derivative with finite-step change and approximation error. 6 practice tasks.
Lesson details and objectives Classify interior critical points and distinguish local tests from global optimization.
Classify interior critical points and distinguish local tests from global optimization. Classify definite and indefinite Hessians and recognize an inconclusive test. Include boundaries and existence conditions when seeking global extrema. 6 practice tasks.
Lesson details and objectives Find regular constrained candidates and inspect exceptional points and global bounds.
Find regular constrained candidates and inspect exceptional points and global bounds. Solve all multiplier candidate cases and check constraint regularity. Use existence and boundary arguments to justify a complete extrema comparison. 6 practice tasks.
Lesson details and objectives Construct area accumulation and distinguish signed integrals from physical mass.
Construct area accumulation and distinguish signed integrals from physical mass. Compute mass and coordinate moments from a nonnegative density. Normalize moments to obtain a center of mass with consistent units. 6 practice tasks.
Lesson details and objectives Reconstruct a region before changing bounds or evaluating an integral.
Reconstruct a region before changing bounds or evaluating an integral. Change integration order while preserving the same region, splitting it when necessary. Justify an iterated-integral interchange under appropriate hypotheses. 6 practice tasks.
Lesson details and objectives Use radius and angle to integrate disk and sector regions with the correct area element.
Use radius and angle to integrate disk and sector regions with the correct area element. Include the polar Jacobian r in area integrals. Distinguish a geometric boundary from multiple covering by a parameterization. 6 practice tasks.
Lesson details and objectives Build volume elements and bounds for solids, then choose coordinates that simplify them.
Build volume elements and bounds for solids, then choose coordinates that simplify them. Use the correct coordinate volume factors and angular conventions. Interpret density, mass and spatial bounds geometrically. 6 practice tasks.
Lesson details and objectives Transform both the region and its local volume scale, checking multiplicity.
Transform both the region and its local volume scale, checking multiplicity. Transform both the integrand and region using the absolute Jacobian. Check injectivity or covering multiplicity and singular boundaries. 6 practice tasks.
Lesson details and objectives Accumulate density along a curve using arc length rather than a coordinate differential.
Accumulate density along a curve using arc length rather than a coordinate differential. Evaluate scalar accumulation against arc length using the speed factor. Explain parameterization and orientation invariance of a scalar line integral. 6 practice tasks.
Lesson details and objectives Compute tangential accumulation and track the effect of direction and parametrization.
Compute tangential accumulation and track the effect of direction and parametrization. Track path orientation and parameter interval consistently. Distinguish vector work from scalar integration against arc length. 6 practice tasks.
Lesson details and objectives Use potentials to evaluate work and distinguish local curl tests from global path independence.
Use potentials to evaluate work and distinguish local curl tests from global path independence. Use endpoint differences when a valid potential exists. Distinguish necessary curl conditions from global path independence on a domain. 6 practice tasks.
Lesson details and objectives Convert circulation to a double integral while checking orientation and holes.
Convert circulation to a double integral while checking orientation and holes. Orient outer and inner boundary curves consistently. Check field regularity and domain hypotheses before applying Green’s theorem. 6 practice tasks.
Lesson details and objectives Construct surface patches, area factors and density integrals without confusing projection with area.
Construct surface patches, area factors and density integrals without confusing projection with area. Use the cross-product magnitude as a surface-area factor. Check regularity and covering when evaluating surface area. 6 practice tasks.
Lesson details and objectives Integrate normal flow through a surface and distinguish signed flux from scalar surface mass.
Integrate normal flow through a surface and distinguish signed flux from scalar surface mass. Calculate flux with a field-normal dot product. Distinguish upward, downward and outward orientations and their sign changes. 6 practice tasks.
Lesson details and objectives Relate circulation to curl flux through an oriented spanning surface.
Relate circulation to curl flux through an oriented spanning surface. Match boundary traversal with surface orientation. Verify smoothness and geometric hypotheses before replacing circulation with flux. 6 practice tasks.
Lesson details and objectives Convert outward flux to volume accumulation and account for missing surfaces and singularities.
Convert outward flux to volume accumulation and account for missing surfaces and singularities. Match every boundary component to its outward normal. Recover an open-surface flux by including and subtracting closure pieces. 6 practice tasks.
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