MATH-14 · Undergraduate · ENGLISH · PLANNED
Build the picture, explain the rule, work an example, then justify it.
Study time is an initial activity estimate, excluding extra reading, revision and extended laboratory or project work. Videos and question bodies have not been produced by adding this outline.
Complete these courses or demonstrate equivalent knowledge. These are recommended prerequisites, not enforced enrolment locks.
YouTube playlist: ARK Mathematics | Multivariable and Vector Calculus
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Sign in to save this learning pathPractical outcome: Map the sensitivity of a two-input design.
Explain surfaces and level sets using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain partial derivatives using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain gradient using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain directional derivatives using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Use the first concepts from this section. Map the sensitivity of a two-input design. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain total differential using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain multivariable chain rule using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain hessian using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Explain lagrange multipliers using a visual and one worked real-world example. Context: Map the sensitivity of a two-input design. End with a prediction or check-for-understanding.
Map the sensitivity of a two-input design. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Surfaces and level sets, Partial derivatives, Gradient and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Estimate the mass of a non-uniform plate.
Explain double integrals using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain iterated integrals using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain polar integration using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain triple integrals using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Use the first concepts from this section. Estimate the mass of a non-uniform plate. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain cylindrical coordinates using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain spherical coordinates using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain jacobians using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Explain change of variables using a visual and one worked real-world example. Context: Estimate the mass of a non-uniform plate. End with a prediction or check-for-understanding.
Estimate the mass of a non-uniform plate. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Double integrals, Iterated integrals, Polar integration and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Compute flow through a simple boundary.
Explain vector fields using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain line integrals using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain conservative fields using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain green's theorem using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Use the first concepts from this section. Compute flow through a simple boundary. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain surface integrals using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain divergence using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain curl using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain stokes' theorem using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Explain divergence theorem using a visual and one worked real-world example. Context: Compute flow through a simple boundary. End with a prediction or check-for-understanding.
Compute flow through a simple boundary. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Vector fields, Line integrals, Conservative fields and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Combine all three sections: Compute flow through a simple boundary. Submit an original worked artifact, evidence or tests, assumptions and a reflection. Rubric: correctness, method, practical usefulness, reproducibility and communication. Self-reviewed, not certified.
6–12 balanced questions sampling every section of Multivariable and Vector Calculus, including interpretation, calculation or reasoning, a misconception and a new application. Passing is 70%; review errors before retrying.
Visual constructions, worked problems and proofs. Guided work leads to an independent attempt, with a diagram, explanation, trace, calculation or tested artifact. Question bodies are generated when opened in your private learning path. Programming submissions are stored as text; the platform does not execute submitted code.
Use original or appropriately licensed datasets. Practical science activities use models or supplied data unless a qualified supervisor has reviewed a physical experiment.
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