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My book has the picture of this situation. Geometrically it's clear that A-cE is a vector perpendicular to E and cE. A = A - cE + cE by parallelogram law. But why do we need A = A - cE + cE to see that A-cE is also perpendicular to cE when (A-cE)· E=0 implies (A-cE)· cE=0? E·E = 1.

Oh yeah, A - cE . E = 0 My bad.

Homework Statement Let E be any unit vector, that is a vector of norm 1. Let c be the component of A along E. We saw that c = A·E. Then A - cE is perpendicular to E, and A = A - cE + cE Then A - cE is also perpendicular to cE ... Homework Equations The Attempt at a Solution I just can't...

Yes. Thank you. Somehow managed to understand this :)

Homework Statement Copy-paste from my textbook: Let S_1 be the sphere of radius 1, centered at the origin. Let a be a number > 0. If X is a point of the sphere S_1, then aX is a point of the sphere of radius a, because ||aX|| = a||X|| = a. In this manner, we get all points of the sphere of...

Homework Statement Use math induction and cases to prove that every integer is even or odd.The Attempt at a Solution Let p(n): n is even or is odd. Let n be an integer. Assume p(n) is true. Then n is even or n is odd. Case 1: Assume n is even. Then n = 2k for some integer k. So n + 1 = 2k +...

Do you mean you'd rather the whole bouquet of redundant theorems/definitions/methods be introduced under an unified umbrella theorem/definition/method? That is, to have only one definition to apply to solve/derive million other definitions/problems and have only handful of those to cover pretty...

I don't understand the difference between Rigorous Calculus books (Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approch", Loomis and Sternberg's "Advanced Calculus", Spivak's "Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus")...

Would Spivak's Calculus and Pugh's Real Math Analysis be sufficient grounding for this text, at least for the "real" part of it? Also, what books and in what order would I need to work my way up from absolute nothing to this book's level in complex analysis? Order of growth towards this...

Awesome! I appreciate you took your time to clarify all this. Thank You, Jbunniii. :cool:

Hi All, I was searching for Analysis course that typically follows Spivak's Calculus. While searching the Net came across this post below and have couple questions if you don't mind: My questions: 1. "Rigorous Advanced Calculus (Loomis and Sternberg)". Is that a single book or two...