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:
-
"Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications" by Tom M. Apostol
-
"Real and Complex Analysis" by Walter Rudin
-
"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.
​
​
​