Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of results which are particularly lengthy (shorter proofs are contained in the notes themselves). The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

"Vector Analysis" (MATH 4317/5317) is no longer a formal class at ETSU. Such a class existed until the late 1990s. The catalog description for that class was: "Topics in vector algebra, vector functions, scalar, and vector fields; line and surface integrals." This is from the 1989-90 catalog. The prerequisites were Linear Algebra (MATH 2010) and Calculus 3 (MATH 2110).

Chapter 1. The Geometry of Euclidean Space.

• Section 1.1. Vectors in Two- and Three-Dimensional Space.
• Section 1.2. The Inner Product, Length, and Distance.
• Section 1.3. Matrices, Determinants, and the Cross Product.
• Section 1.4. Cylindrical and Spherical Coordinates.
• Section 1.5. n-Dimensional Euclidean Space.

Chapter 2. Differentiation.

• Section 2.1. The Geometry of Real-Valued Functions.
• Section 2.2. Limits and Continuity.
• Section 2.3. Differentiation.
• Section 2.4. Introduction to Paths and Curves.
• Section 2.5. Properties of the Derivative.
• Section 2.6. Gradients and Directional Derivatives.

Chapter 3. Higher-Order Derivatives; Maxima and Minima.

• Section 3.1. Iterated Partial Derivatives.
• Section 3.2. Taylor's Theorem.
• Section 3.3. Extrema of Real-Valued Functions.
• Section 3.4. Constrained Extrema and Lagrange Multipliers.
• Section 3.5. The Implicit Function Theorem. Section 3.5 notes
• Supplement. Examples and Proofs from Section 3.5. Beamer file of Section 3.5 Examples and Proofs
• Supplement. Printouts of Examples and Proofs from Section 3.5. Print file of Section 3.5 Examples and Proofs

Chapter 4. Vector-Valued Functions.

• Section 4.1. Acceleration and Newton's Second Law.
• Section 4.2. Arc Length.
• Section 4.3. Vector Fields.
• Section 4.4. Divergence and Curl.

Chapter 5. Double and Triple Integrals.

• Section 5.1. Introduction.
• Section 5.2. The Double Integral Over a Rectangle.
• Section 5.3. The Double Integral Over More General Regions.
• Section 5.4. Changing the Order of Integration.
• Section 5.5. The Triple Integral.

Chapter 6. The Change of Variables Formula and Applications of Integration.

• Section 6.1. The Geometry of Maps from ℝ² to ℝ².
• Section 6.2. The Change of Variables Theorem.
• Section 6.3. Applications.
• Section 6.4. Improper Integrals.

Chapter 7. Integrals Over Paths and Surfaces.

• Section 7.1. The Path Integral.
• Section 7.2. Line Integrals.
• Section 7.3. Parametrized Surfaces.
• Section 7.4. Area of a Surface.
• Section 7.5. Integrals of Scalar Functions Over Surfaces.
• Section 7.6. Surface Integrals of Vector Functions.
• Section 7.7. Applications to Differential Geometry, Physics, and Forms of Life .

Chapter 8. The Integral Theorems of Vector Analysis.

• Section 8.1. Green's Theorem.
• Section 8.2. Stoke's Theorem.
• Section 8.3. Conservative Fields.
• Section 8.4. Gauss' Theorem.
• Section 8.5. Differential Forms.

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