MATH2010 Advanced Calculus I
Develop multivariable calculus from Euclidean geometry and limits to continuity, differentiability, Jacobian matrices, and the chain rule.
1.1 Vectors, inner products, and norms
Choose a method, work through an original practice paper, and use complete marking to return to the step that needs work.
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Chapter 1
Euclidean geometry and coordinates
Build the language of vectors, distances, lines, planes, coordinate systems, and surfaces used throughout multivariable calculus.
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1.1 Vectors, inner products, and norms
Use coordinates, inner products, norms, and equality cases to connect vector calculations with Euclidean geometry.
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1.2 Lines, planes, areas, and volumes
Derive equations of lines and planes, then use projection and vector products to calculate distances, areas, and volumes.
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1.3 Coordinates, curves, and surfaces
Translate between Cartesian, polar, cylindrical, and spherical coordinates and interpret curves, level sets, and quadric surfaces.
Chapter 2
Point sets, limits, and continuity
Describe domains precisely, control every approach to a point, and connect continuity with compactness and existence results.
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2.1 Domains and point sets
Determine natural domains and use balls, interior points, boundaries, open sets, and closed sets to describe them precisely.
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2.2 Sequences and multivariable limits
Prove limits with uniform estimates and the sequence criterion, and distinguish path tests from proofs of existence.
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2.3 Continuity and compactness
Use continuity, compactness, and convergent subsequences to establish existence of extrema and uniform control.
Chapter 3
Multivariable differentiation
Pass from coordinate derivatives to a linear map, then compose derivatives and interpret rates along curves and directions.
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3.1 Partial derivatives and gradients
Compute partial and mixed derivatives, organize first-order information in the gradient, and state the regularity needed to interchange derivatives.
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3.2 Differentiability and linearization
Define differentiability through a linear approximation with a small relative error, and derive tangent planes, differentials, and the C1 criterion.
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3.3 Jacobians, chain rule, and directional derivatives
Apply Jacobian matrices and the chain rule to compositions, curves, directional rates, and steepest ascent.