Prove that the differential operator is linear, i. D()\f + pg) = ADU) + pD(g). This is similar to exercise 2 in section 6.1, page 330. Combining the...

Multivariable analysis course. 5 fairly standard questions and 1 proof question.

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1. Prove that the differential operator is linear, i.e. D()\f + pg) = ADU) +pD(g). This is similar to exercise 2 in section 6.1, page 330. 2. Combining the example we talked about on lecture 4, Jan. 25. Do theproblem 3, page 330, exercise 3. In case some of you do not have the textbook,we Itate it as below. Let A := {(m,y) E R2|0 S a: S 1,y = 0}. Prove the uniqueness of thederivatives result (refer to my note posted on Jan. 25.) iI falls for this A. 3. Problem 4, page 330. Let f : IR” —> Rm and suppose there is a constantM such that a: E R”, Hf(:z:)H S MHm||2. Prove f is differentiable at 3:0 = O andthat D 1' ($0) = O. This problem should be compared to problem 3, page 334,exercises for section 6.2. But you don’t need to write down the solution for the2nd problem. 4. Problem 5, page 330. If f : IR —) R is difi'erentiable and |f(:z:)| 5 |:I:|, mustD f (0) = 0? 5. Problem 2, page 334. Let f : R3 —> IR, (93,31, 2) —> 632+92+22. ComputeDU) and grad f. 6. Let f : IR” —> Rm be a constant function. Show that D f (0) = O for any330 E R”. (Note that the 1st 0 is in R", while the 2nd 0 is the zero matrix orthe zero element in EUR”, Rm). )