Exercise 1.6.3. Fix a differentiable function f: R2 + R, and fit a point p € R. For any vector v € R2, then the directio

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Exercise 1 6 3 Fix A Differentiable Function F R2 R And Fit A Point P R For Any Vector V R2 Then The Directio 1 (33.08 KiB) Viewed 29 times

Exercise 1.6.3. Fix a differentiable function f: R2 + R, and fit a point p € R. For any vector v € R2, then the directional derivative off at p in the direction of v is defined as of := lime-0 p+w)-1(P) Show that the map :0 HV) is a dual vector in (R2), i.e., a row vector. Also, what are its "coordinates" under the standard dual basis? (Remark: In calculus, we write vs as a column vector for historical reasons. By all means, from a mathematical perspective, the correct way to write f is to write it as a rou vector, as illustrated in this problem. (But don't annoy your calculus teachers though.... In your calculus class, you use whatever notation your calculus teacher told you.) (Extra Remark: If we use row vector, then the evaluation of VS at v is purely linear, and no inner product structure is needed, which is sweet. But if we HAVE TO write Vf as a column vector (for historical reason), then we would have to do a dot product between Vf and v, which now requires an inner product structure. That is an unnecessary dependence on an extra structure that actually should have no influence.)