MATH2066: Introductory Real Analysis — Taylor supplement
Notes on higher derivatives, multi-index Taylor coefficients and rigorous remainder estimates.
All-order multivariable Taylor theorem
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Chapter 1
All-order multivariable Taylor theorem
Multi-index coefficients, integral and Lagrange remainders, and the local Peano theorem, with full proofs and solved practice.
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All-order multivariable Taylor theorem
Multi-index coefficients, integral and Lagrange remainders, and the local Peano theorem, with full proofs and solved practice.
Chapter 1
Multivariable differentiability and mean value estimates
Several directional rates do not automatically assemble into a good linear approximation. The central analytical question is whether one linear map controls the error for every sufficiently small displacement, not just along a chosen line. This distinction is the foundation for chain rules, Taylor estimates and multivariable optimization.
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Multivariable differentiability and mean value estimates
Several directional rates do not automatically assemble into a good linear approximation. The central analytical question is whether one linear map controls the error for every sufficiently small displacement, not just along a chosen line. This distinction is the foundation for chain rules, Taylor estimates and multivariable optimization.
Chapter 2
Taylor expansion and convexity
A derivative predicts the first-order change. A Hessian predicts how that rate changes with direction. Taylor's theorem makes the approximation quantitative, while convexity turns local derivative information into global comparisons. The hypotheses and the order of the remainder are part of each result.
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Taylor expansion and convexity
A derivative predicts the first-order change. A Hessian predicts how that rate changes with direction. Taylor's theorem makes the approximation quantitative, while convexity turns local derivative information into global comparisons. The hypotheses and the order of the remainder are part of each result.
Chapter 3
Multiple Riemann integration and Fubini
An iterated integral is a sequence of one-variable integrals. A multiple integral is a limit of sums over small cells. Their equality requires a theorem. We establish it first for continuous functions on rectangles, where the hypotheses are transparent, and then use the result to compute and interpret region integrals.
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Multiple Riemann integration and Fubini
An iterated integral is a sequence of one-variable integrals. A multiple integral is a limit of sums over small cells. Their equality requires a theorem. We establish it first for continuous functions on rectangles, where the hypotheses are transparent, and then use the result to compute and interpret region integrals.
Chapter 4
Uniform convergence and interchange of limits
A sequence of good functions can have a badly behaved limit. The issue is whether convergence becomes accurate at one common stage for all points. Uniform convergence supplies that control, but it does not justify every operation equally: integration is more forgiving than differentiation.
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Uniform convergence and interchange of limits
A sequence of good functions can have a badly behaved limit. The issue is whether convergence becomes accurate at one common stage for all points. Uniform convergence supplies that control, but it does not justify every operation equally: integration is more forgiving than differentiation.
Chapter 5
Power series and real analyticity
A Taylor polynomial approximates a function to finite order. An analytic representation makes an infinite series equal the function on a neighborhood. Smoothness alone does not guarantee that equality. Power-series convergence and remainder control explain the difference.
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Power series and real analyticity
A Taylor polynomial approximates a function to finite order. An analytic representation makes an infinite series equal the function on a neighborhood. Smoothness alone does not guarantee that equality. Power-series convergence and remainder control explain the difference.
Chapter 6
Riemann slices and noncontinuous Fubini
Continuity on a rectangle is a clean sufficient condition for changing the order of integration, but many useful functions have jumps on curves. We now prove a broader bounded Riemann slicing theorem, explain the separate role of slice existence, and examine an improper example where reversing order changes the answer.
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Riemann slices and noncontinuous Fubini
Continuity on a rectangle is a clean sufficient condition for changing the order of integration, but many useful functions have jumps on curves. We now prove a broader bounded Riemann slicing theorem, explain the separate role of slice existence, and examine an improper example where reversing order changes the answer.
Chapter 7
Higher derivatives and multivariable analyticity
A several-variable Taylor formula is not obtained by writing unrelated one-variable expansions in each coordinate. Mixed derivatives occur because a displacement changes all coordinates at once. Multilinear derivatives organize those interactions, and a remainder estimate tells us whether the infinite Taylor expansion equals the function.
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Higher derivatives and multivariable analyticity
A several-variable Taylor formula is not obtained by writing unrelated one-variable expansions in each coordinate. Mixed derivatives occur because a displacement changes all coordinates at once. Multilinear derivatives organize those interactions, and a remainder estimate tells us whether the infinite Taylor expansion equals the function.
Chapter 8
Riemann integration the fundamental theorem and improper integrals
An integral can exist even when its integrand has jumps, and an antiderivative can recover that integral under hypotheses weaker than continuity. An improper integral introduces a different limit, at an infinite endpoint or a singularity. We keep these constructions separate so that the fundamental theorem is applied only where it is justified.
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Riemann integration the fundamental theorem and improper integrals
An integral can exist even when its integrand has jumps, and an antiderivative can recover that integral under hypotheses weaker than continuity. An improper integral introduces a different limit, at an infinite endpoint or a singularity. We keep these constructions separate so that the fundamental theorem is applied only where it is justified.
Chapter 9
Double series and Fubini reordering
Summing each row and then summing the rows is a different limiting process from summing columns or expanding square rectangles. These processes agree under nonnegativity or absolute summability. Conditional examples show why a double-integral chapter cannot stand in for the separate theorem about double series.
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Double series and Fubini reordering
Summing each row and then summing the rows is a different limiting process from summing columns or expanding square rectangles. These processes agree under nonnegativity or absolute summability. Conditional examples show why a double-integral chapter cannot stand in for the separate theorem about double series.
Chapter 10
Rigorous exponential logarithmic and trigonometric functions
A familiar graph is not a mathematical definition. We construct the elementary functions from convergent series, prove their algebraic and differential laws, and define their inverse branches only after proving the necessary ranges. This closes a different gap from applying a Taylor formula to a function whose existence and properties were already assumed.
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Rigorous exponential logarithmic and trigonometric functions
A familiar graph is not a mathematical definition. We construct the elementary functions from convergent series, prove their algebraic and differential laws, and define their inverse branches only after proving the necessary ranges. This closes a different gap from applying a Taylor formula to a function whose existence and properties were already assumed.
Chapter 11
One variable mean value and Taylor proofs
The segment arguments in several variables depend on one-variable theorems. Here those foundations are proved within MATH2066 rather than counted from another course. The difference between differentiability, continuous differentiability and interval-wide hypotheses remains visible throughout.
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One variable mean value and Taylor proofs
The segment arguments in several variables depend on one-variable theorems. Here those foundations are proved within MATH2066 rather than counted from another course. The difference between differentiability, continuous differentiability and interval-wide hypotheses remains visible throughout.