MATH 5720: Vector Calculus&Complex Variables

List of topics

Chapter 9.Vector Differential Calculus: Vector Algebra in R2 an R3. Inner Product (Dot Product). Vector Product (Cross Product) Vector and Scalar Functions and Fields. Derivatives. Curves. Tangents. Arc Length. Curves in mechanics. Velocity and Acceleration. Gradient of a Scalar Field. Directional Derivative. Divergence of a Vector Field. Curl of a Vector Field.

Chapter 10.Vector Integral Calculus: Line Integrals. Line Integrals Independent of Path. Double Integrals. Green's Theorem in the Plane. Surfaces for Surface Integrals. Surface Integrals. Triple Integrals. Divergence Theorem of Gauss. Applications of the Divergence Theorem. Stokes's Theorem.

Chapter 13.Complex numbers and functions: Complex Numbers. Complex Plane. Polar Form of Complex  Numbers.  Powers and Roots. Derivative. Analytic Function. Cauchy-Riemann Equations. Laplace's Equation. Geometry of Analytic Functions: Conformal Mapping. Exponential Function. Trigonometric Functions, Hyperbolic Functions. Logarithm. General Power. Linear Fractional Transformations.

Chapter14.Complex Integration: Line Integral in the Complex Plane. Cauchy's Integral Theorem. Cauchy's Integral Formula. Derivatives of Analytic Functions.

Chapter 15.Power Series, Taylor Series: Sequences, Series, Convergence Tests. Power Series. Functions Given by Power Series. Taylor Series and Maclaurin Series.

Chapter 16.Laurent Series, Residue Integration: Laurent Series. Singularities and Zeros. Infinity. Residue Integration Method. Evaluation of Real Integrals.

Chapter 17.Conformal Mappings: Geometry of analytical functions. Linear Fractional

Chapter 18.Applications to Potential Theory: Electrostatic Fields. Use of Conformal Mapping. Heat Problems. Fluid Flow. Poisson's Integral Formula. General Properties of Harmonic Functions.