      On this webpage you will find my solutions to the sixth edition of "Vector Calculus" by Jerrold E. Marsden and Anthony Tromba. Here is a link to the book's page on amazon.com. If you find my work useful, please consider making a donation. Thank you.

• 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 (Optional)
• 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 (Optional)
• 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 Fields
• Section 7.7: Applications of Differential Geometry, Physics, and Forms of Life

• Chapter 8: The Integral Theorems of Vector Analysis
• Section 8.1: Green's Theorem
• Section 8.2: Stokes' Theorem
• Section 8.3: Conservative Fields
• Section 8.4: Gauss' Theorem
• Section 8.5: Differential Forms
Section 1.3 Section 1.4
Exercise 1 Exercise 9 Exercise 17 Exercise 25 Exercise 33 Exercise 41 Exercise 1 Exercise 9 Exercise 17
Exercise 2 Exercise 10 Exercise 18 Exercise 26 Exercise 34 Exercise 42 Exercise 2 Exercise 10 Exercise 18
Exercise 3 Exercise 11 Exercise 19 Exercise 27 Exercise 35 Exercise 43 Exercise 3 Exercise 11 Exercise 19
Exercise 4 Exercise 12 Exercise 20 Exercise 28 Exercise 36 Exercise 44 Exercise 4 Exercise 12 Exercise 20
Exercise 5 Exercise 13 Exercise 21 Exercise 29 Exercise 37 Exercise 45 Exercise 5 Exercise 13 Exercise 21
Exercise 6 Exercise 14 Exercise 22 Exercise 30 Exercise 38 Exercise 46 Exercise 6 Exercise 14 Exercise 22
Exercise 7 Exercise 15 Exercise 23 Exercise 31 Exercise 39 Exercise 7 Exercise 15
Exercise 8 Exercise 16 Exercise 24 Exercise 32 Exercise 40 Exercise 8 Exercise 16
Section 1.5 Chapter 1 Review
Exercise 1 Exercise 11 Exercise 21 Exercise 1 Exercise 11 Exercise 21 Exercise 31 Exercise 41 Exercise 51
Exercise 2 Exercise 12 Exercise 22 Exercise 2 Exercise 12 Exercise 22 Exercise 32 Exercise 42 Exercise 52
Exercise 3 Exercise 13 Exercise 23 Exercise 3 Exercise 13 Exercise 23 Exercise 33 Exercise 43 Exercise 53
Exercise 4 Exercise 14 Exercise 24 Exercise 4 Exercise 14 Exercise 24 Exercise 34 Exercise 44 Exercise 54
Exercise 5 Exercise 15 Exercise 5 Exercise 15 Exercise 25 Exercise 35 Exercise 45 Exercise 55
Exercise 6 Exercise 16 Exercise 6 Exercise 16 Exercise 26 Exercise 36 Exercise 46 Exercise 56
Exercise 7 Exercise 17 Exercise 7 Exercise 17 Exercise 27 Exercise 37 Exercise 47 Exercise 57
Exercise 8 Exercise 18 Exercise 8 Exercise 18 Exercise 28 Exercise 38 Exercise 48
Exercise 9 Exercise 19 Exercise 9 Exercise 19 Exercise 29 Exercise 39 Exercise 49
Exercise 10 Exercise 20 Exercise 10 Exercise 20 Exercise 30 Exercise 40 Exercise 50