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Advanced Calculus

Course Outline

I. Review of Basic Calculus

A. Functions, Limits, and Continuity B. Differentiation C. Integration

II. Multivariable Calculus

A. Functions of Several Variables B. Partial Derivatives C. Multiple Integrals D. Vector Calculus: Divergence and Curl, Line and Surface Integrals

III. Infinite Series

A. Convergence of Sequences and Series B. Power Series C. Taylor and Maclaurin Series

 

IV. Differential Equations

A. First-Order Differential Equations B. Second-Order Differential Equations C. Systems of Differential Equations D. Laplace Transforms

V. Real Analysis

A. Sets and Functions in R^n B. Sequences and Series in R^n C. Compactness and Connectedness D. Continuity and Differentiability in R^n

VI. Complex Analysis

A. Complex Numbers and Functions B. Complex Differentiation and the Cauchy-Riemann Equations C. Complex Integration and Cauchy's Theorem D. Power Series, Taylor Series, and Laurent Series E. Residue Theorem and Contour Integration

VII. Metric Spaces

A. Definition and Examples B. Open, Closed, and Compact Sets C. Continuity and Uniform Continuity in Metric Spaces D. Complete Metric Spaces and Fixed Point Theorems

VIII. Functional Analysis

A. Normed Spaces, Banach Spaces B. Inner Product Spaces, Hilbert Spaces C. Linear Functionals and the Hahn-Banach Theorems D. Compact Operators and the Spectral Theorems

IX. Special Topics

A. Fourier Analysis and Partial Differential Equations B. Calculus of Variations C. Introduction to Measure Theory and Lebesgue Integration

Textbooks:

  1. "Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications" by Tom M. Apostol

  2. "Real and Complex Analysis" by Walter Rudin

  3. "Functional Analysis" by Michael Reed and Barry Simon.

I. Review of Basic Calculus

 

We will be reviewing some essential concepts from basic calculus: functions, limits, continuity, differentiation, and integration. These concepts form the building blocks for this course and will provide the basis for our exploration of more advanced topics.

A. Functions, Limits, and Continuity

A function is a rule that assigns a unique output for each input. For example, the function f(x) = x^2 assigns the output x^2 for each input x.

The limit of a function describes the behavior of the function as the input approaches a particular value. For example, as x approaches 0, the limit of the function f(x) = x^2 is 0, because as we make x closer and closer to 0, the value of x^2 becomes closer and closer to 0.

A function is continuous at a point if the function's value at that point is equal to the limit of the function as x approaches that point. For instance, the function f(x) = x^2 is continuous at x = 1, because the limit of f(x) as x approaches 1 is 1, which is the same as the function's value at x = 1.

B. Differentiation

Differentiation is the process of finding a function's derivative, which measures how the function changes as its input changes. The derivative of a function at a particular point is the slope of the line tangent to the function's graph at that point.

For example, the derivative of f(x) = x^2 is f'(x) = 2x. This tells us that, for any value of x, the function f(x) = x^2 changes at a rate of 2x.

C. Integration

Integration is, in a sense, the reverse of differentiation. It is the process of finding a function's integral, which represents the area under the curve of the function's graph.

For example, the integral of f(x) = x^2 from 0 to 1 is 1/3, because the area under the curve y = x^2 from x = 0 to x = 1 is 1/3.

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