Intereting Posts

Multiple choice question – number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$
A quick question on transcendence
Inverse Image of Maximal Ideals
Is Cross Product Defined on Vector Space?
Groups having at most one subgroup of any given finite index
Simply formulated but very hard problem about certain polynomial
$\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3}$ using complex analysis
Existence of an element of given orders at finitely many prime ideals of a Dedekind domain
Why is an empty function considered a function?
Proving $\displaystyle\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$
Explain a surprisingly simple optimization result
How would you explain why “e” is important? (And when it applies?)
On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$
Proj construction and ample dualizing sheaf
Polynomials with rational zeros

I’m looking for a book on differential geometry or differential topology that is comprehensive and reads at the level of someone with engineering background (i.e. Boyce’s ODE, Stewart’s Calculus, Axler’s Linear algebra). The book should motivate the idea of manifold as it is used in physics and engineering and move up to stuff like vector bundle, wedge products, Poincaré–Hopf theorem and maybe at the very very end some Clifford algebra (helpful with application to electromagnetism or general relativity).

The book I’ve surveyed which includes Janich’s Intro to Differential Topology, Isham’s Differential Geometry for Physicists, Differential Manifold by Serge Lang, Introduction to Manifolds by Tu L.W. unfortunately all reads like books written by mathematicians for mathematicians and has a dearth of physical examples and visual aids. Tu L.W.’s Intro to Manifold is surprisingly soft handed and perhaps would be good for a first book. The book nonetheless lacks motivating examples and illuminating graphs.

Can someone who has taught differential geometry to engineers or physicists or perhaps know a good introductory book on this subject recommend a book that covers about half semester worth of undergrad?

- What is a $k$-form on $^k$ in Spivak's Calculus on Manifolds?
- Volume of spheres in higher dimensions?
- Turning higher spheres inside out
- Does the curvature determine the metric?
- The Affine Property of Connections on Vector Bundles
- Spin manifold and the second Stiefel-Whitney class

Thanks!

- Foundations of logic
- Is it possible to elementarily parametrize a circle without using trigonometric functions?
- Advice about taking mathematical analysis class
- Relating the Künneth Formula to the Leray-Hirsch Theorem
- A good reference to begin analytic number theory
- Why do people interchange between $\int$ and $\sum$ so easily?
- Euler's errors?
- Closed-form Expression of the Partition Function $p(n)$
- Five squares in a box.
- Good calculus exercises/problems?

Gauge Fields, Knots and Gravity by Baez and Muniain. The mathematical prerequisites are few and the book gives a great amount of physical intuition without being too sloppy. You will not find many ‘engineering’ applications of manifolds here, but electromagnetism and other gauge theories are treated extensively.

Try The Geometry of Physics. I think, however, that this will not be totally satisfying to you, because it is very advanced.

- Equilateral triangle in a circle
- Uniform integrability question
- Tensor product between quaternions and complex numbers.
- Number of ways of forming 4 letter words using the letters of the word RAMANA
- Orthogonality of projections on a Hilbert space
- Produce an explicit bijection between rationals and naturals?
- message plus response probability problem
- Splitting a sandwich and not feeling deceived
- Neutral element in $\hom_C(A, B)$
- Proof by induction: $\sum\limits_{i=1}^{n} \frac{1}{n+i} = \sum\limits_{i=1}^{n} \left(\frac{1}{2i-1} – \frac{1}{2i}\right)$
- Closed space curves of constant curvature
- How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent
- On the derivative of a Heaviside step function being proportional to the Dirac delta function
- When the product of dice rolls yields a square
- $\gcd(a,b) = \gcd(a + b, \mathrm{lcm})$