Advanced Calculus: A Geometric View - Errata
This page contains corrections and clarifications for the textbook Advanced Calculus: A Geometric View. While every effort is made to ensure accuracy in mathematical publications, errors can occur. This errata sheet documents known errors and corrections that will be incorporated into future printings and editions.
We are grateful to the many students and instructors who have taken the time to point out errors and share suggestions for improvement. Their contributions have been invaluable in enhancing the quality of this text.
About This Text
Advanced Calculus: A Geometric View presents calculus with a special emphasis on geometric understanding. Rather than focusing solely on algebraic manipulation, the text develops multivariable calculus concepts through visual and geometric interpretations. This approach helps students develop intuition about limits, derivatives, integrals, and other calculus concepts in higher dimensions.
Errata by Chapter
The definition of a differentiable function at a point incorrectly states that the partial derivatives must be continuous at that point.
Correction: A function f: is differentiable at a point a if there exists a linear map L: such that f(a + h) = f(a) + L(h) + r(h) where r(h)/||h|| 0 as h 0. The existence of continuous partial derivatives is a sufficient condition for differentiability but not the definition itself.
In the parametric curve representation, the parameter values for points A and B are labeled incorrectly.
Correction: Point A is at parameter value t = 0 and point B at t = 1, not the reverse as labeled in the figure.
The domain of integration is incorrectly specified in the problem statement.
Correction: The integration should be over the region R = {(x,y) | 0 x 1, x y x} rather than as printed.
The condition for the Inverse Function Theorem is missing the requirement that the derivative map be invertible.
Correction: The theorem should state: "If f: is continuously differentiable in an open set containing a, and if Df(a) is invertible (i.e., its Jacobian determinant is nonzero at a), then there exists an open neighborhood U of a such that f maps U bijectively onto an open set V in the codomain..."
The second line of the derivation incorrectly swaps the integration order without adjusting the limits accordingly.
Correction: When swapping the order, the new limits should be 0 y 4 and 0 x y, not as originally shown.
Minor Corrections
| Page | Issue | Correction |
|---|---|---|
| 23 | "calculus" misspelled as "calclulus" | "calculus" |
| 67 | Extra parenthesis in equation 3.2.4 | Remove the extra closing parenthesis |
| 134 | Referenced equation 4.5 does not exist | See Equation 4.6 |
| 201 | "Taylor" misspelled as "Tayor" | "Taylor" |
| 245 | Missing "is" in sentence | "The function continuous" should be "The function is continuous" |
| 278 | Figure caption refers to wrong equation | "Figure 7.3" should reference Equation 7.8 not 7.6 |
Additional Clarifications
Several readers have found the definition of compact sets confusing due to the terminology used.
Clarification: A compact set in is a set that is closed and bounded. The discussion of open covers, while mathematically correct, may be omitted on first reading. For applied purposes in calculus, the "closed and bounded" characterization is typically sufficient.
The transition from equation (6.8.2) to (6.8.3) has caused confusion for some students.
Clarification: An intermediate step showing the substitution of the parameterization into the line integral would be helpful. Let x = 2cos(t), y = 2sin(t), then dr = (-2sin(t)dt, 2cos(t)dt). Substituting these into the integral yields equation (6.8.3).
The proof sketch omits the treatment of the boundary case, which makes understanding the complete proof challenging.
Clarification: To complete the proof, one must also consider points on the boundary of the domain. The approach is similar to the interior case but using half-neighborhoods rather than full neighborhoods. Alternatively, extend the definition of f to a larger domain using reflection, apply the theorem, and then restrict back.
Notation Clarifications
The notation ||v|| for a vector v's norm is sometimes used interchangeably with |v| without explanation.
Clarification: Throughout this text, ||v|| and |v| both denote the Euclidean norm (length) of the vector v. In , for v = (v, v, ..., v), ||v|| = |v| = (v + v + ... + v). This dual notation is common in mathematical literature, though we aim for consistency within each chapter where possible.
The notation Df(a) is sometimes used for the derivative matrix and sometimes for the linear transformation, which can be confusing.
Clarification: In Chapter 5, we use Df(a) to represent both the derivative matrix and the linear transformation it represents. Technically, Df(a) is the linear transformation, and the matrix representation depends on the choice of basis. However, since we work in the standard basis until Chapter 8, this distinction rarely matters for calculations.
Reporting New Errata
We continue to welcome feedback about errors in the text. When reporting potential errata, please include:
- The page number
- The location on the page (paragraph, equation number, figure, etc.)
- The issue as it appears in the text
- The suggested correction
- Your name (optional, if you would like to be credited)
Please send errata reports to advancedcalculus.errata@mathpublishing.com
This errata page was last updated on June 15, 2023.
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