By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Booklet via Michael Spivak, Spivak, Michael

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**Extra resources for A Comprehensive Introduction to Differential Geometry**

**Example text**

36. Expand and simplify the following: 1. [(x − y) dx + (x + y) dy + z dz] ∧ [(x − y) dx + (x + y) dy]. 2. (2dx + 3dy) ∧ (dx − dz) ∧ (dx + dy + dz). 1 Families of forms Let us now go back to the example in Chapter 1. In the last section of that chapter, we showed that the integral of a function, f : R3 → R, over a surface parameterized by φ : R ⊂ R2 → R3 is f (φ(r, θ))Area ∂φ ∂φ (r, θ), (r, θ) dr dθ. ∂r ∂θ R This gave one motivation for studying diﬀerential forms. We wanted to generalize this integral by considering functions other than “Area(·, ·)” that eat pairs of vectors and return numbers.

Such a vector is easy to ∇f ﬁnd: U = |∇f | . Now we compute this slope: ∇U f = ∇f · U ∇f = ∇f · |∇f | 1 = (∇f · ∇f ) |∇f | 1 |∇f |2 = |∇f | = |∇f |. Hence, the magnitude of the gradient vector represents the largest slope of a tangent line through a particular point. 9. Let f (x, y) = sin(xy 2 ). Calculate the directional derivative of f (x, y) at √ √ the point π4 , 1 , in the direction of 22 , 22 . 10. Let f (x, y) = xy 2 . 1. Compute ∇f . 2. Use your answer to the previous question to compute ∇ 1,5 f (2, 3).

Proof. The proof of the above lemma relies heavily on the fact that 2-forms which are the product of 1-forms are very ﬂexible. The 2-form α ∧ β takes pairs of vectors, projects them onto the plane spanned by the vectors α and β , and computes the area of the resulting parallelogram times the area of the parallelogram spanned by α and β . Note that for every nonzero scalar c, the area of the parallelogram spanned by α and β is the same as the area of the parallelogram spanned by c α and 1/c β . ) The important point here is that we can scale one of the 1-forms as much as we want at the expense of the other and get the same 2-form as a product.

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