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?