By David Bachman
The smooth topic of differential varieties subsumes classical vector calculus. this article offers differential types from a geometrical point of view available on the complex undergraduate point. the writer methods the topic with the concept complicated recommendations will be equipped up by way of analogy from less complicated situations, which, being inherently geometric, frequently should be most sensible understood visually.
Each new idea is gifted with a common photograph that scholars can simply grab; algebraic homes then stick with. This allows the advance of differential varieties with out assuming a historical past in linear algebra. through the textual content, emphasis is put on purposes in three dimensions, yet all definitions are given in order to be simply generalized to better dimensions.
The moment variation features a thoroughly new bankruptcy on differential geometry, in addition to different new sections, new workouts and new examples. extra ideas to chose routines have additionally been integrated. The paintings is acceptable to be used because the basic textbook for a sophomore-level classification in vector calculus, in addition to for extra upper-level classes in differential topology and differential geometry.
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Extra info for A Geometric Approach to Differential Forms
Example text
3. 3dx ∧ dy + dy ∧ dx + dx ∧ dz. 4. dx ∧ dy + 3dz ∧ dy + 4dx ∧ dz. 4 2-Forms on Tp R3 (optional) This text is about differential n-forms on Rm . For most of it, we keep n, m ≤ 3 so that everything we do can be easily visualized. However, very little is special about these dimensions. Everything we do is presented so that it can easily generalize to higher dimensions. In this section and the next we break from this philosophy and present some special results when the dimensions involved are 3 or 4.
11. (ω + ν) ∧ ψ = ω ∧ ψ + ν ∧ ψ (∧ is distributive). There is another way to interpret the action of ω ∧ ν which is much more geometric. First, let ω = a dx + b dy be a 1-form on Tp R2 . Then we let ω be the vector a, b . 12. Let ω and ν be 1-forms on Tp R2 . Show that ω ∧ ν(V1 , V2 ) is the area of the parallelogram spanned by V1 and V2 , times the area of the parallelogram spanned by ω and ν . 13. Use the previous problem to show that if ω and ν are 1-forms on R2 such that ω ∧ ν = 0, then there is a constant c such that ω = cν.
If V is a vector, then we let V −1 be the corresponding 1-form. We now prove two lemmas. Lemma 1. If α and β are 1-forms on Tp R3 and V is a vector in the plane spanned by α and β , then there is a vector, W , in this plane such that α ∧ β = V −1 ∧ W −1 . 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 .