By David Bachman

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3.

One reason is that the derivative is actually a vector. If φ(t) = (f (t), g(t)), then d dφ = (f (t), g(t)) = f (t), g (t) . dt dt This vector has important geometric signiﬁcance. The slope of a line containing this vector when t = t0 is the same as the slope of the line tangent to the curve at the point φ(t0 ). The magnitude (length) of this vector gives one a concept of the speed of the point φ(t) as t is increases through t0 . 1). 5. Let φ(t) = (cos t, sin t) (where 0 ≤ t ≤ π ) and ψ(t) = (t, 1 − t 2 ) (where −1 ≤ t ≤ 1) be parameterizations of the top half of the√unit√circle.

We conclude Evaluating ω∧ν on the pair of vectors (V1 , V2 ) gives the area of parallelogram spanned by V1 and V2 projected onto the plane containing the vectors ω and ν , and multiplied by the area of the parallelogram spanned by ω and ν . CAUTION: While every 1-form can be thought of as projected length not every 2-form can be thought of as projected area. The only 2-forms for which this interpretation is valid are those that are the product of 1-forms. 18. Let’s pause for a moment to look at a particularly simple 2-form on Tp R3 , dx ∧ dy.