What sort of answer does Spivak expect to Problem 2-13(d) from Calculus on Manifolds?

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Problem 2-13 from Spivak's Calculus on Manifolds asks the following:

Problem 2-13. Define $IP : mathbbR^n times mathbbR^n to mathbbR$ by $IP(x,y) = langle x, y rangle$.

1. Find $D(IP)(a,b)$ and $(IP)'(a,b)$.


2. If $f,g : mathbbR to mathbbR^n$ are differentiable and $h : mathbbR to mathbbR$ is defined by $h(t) = langle f(t), g(t) rangle$, show that $$h'(a) = langle f'(a)^T, g(a) rangle + langle f(a), g'(a)^T rangle.$$


3. If $f : mathbbR to mathbbR$ is differentiable and $|f(t)| = 1$ for all $t$, show that $langle f'(t)^T, f(t) rangle = 0$.


4. Exhibit a differentiable function $f : mathbbR to mathbbR$ such that the function $|f|$ defined by $|f|(t) = |f(t)|$ is not differentiable.

What I find weird is that part (4) of the problem seems quite unrelated to the previous three parts of the question. The first three parts are specifically about the inner product as a multilinear function and the properties it and its derivative have, whereas for part (4) I can just take $f(x) = x$ and be done with it.

Typically, Spivak's problems are set up in a way to motivate some deeper idea. It is possible that he has slipped up here (after all, Calculus on Manifolds has numerous typos as well). But, I'd still like to ask,

Is there a way to look at part (4) of this problem in such a way that it connects to the inner product (or more precisely, to the previous three parts of the problem)?

real-analysis multivariable-calculus

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asked 5 hours ago

Brahadeesh

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Problem 2-13 from Spivak's Calculus on Manifolds asks the following:

Problem 2-13. Define $IP : mathbbR^n times mathbbR^n to mathbbR$ by $IP(x,y) = langle x, y rangle$.

1. Find $D(IP)(a,b)$ and $(IP)'(a,b)$.


2. If $f,g : mathbbR to mathbbR^n$ are differentiable and $h : mathbbR to mathbbR$ is defined by $h(t) = langle f(t), g(t) rangle$, show that $$h'(a) = langle f'(a)^T, g(a) rangle + langle f(a), g'(a)^T rangle.$$


3. If $f : mathbbR to mathbbR$ is differentiable and $|f(t)| = 1$ for all $t$, show that $langle f'(t)^T, f(t) rangle = 0$.


4. Exhibit a differentiable function $f : mathbbR to mathbbR$ such that the function $|f|$ defined by $|f|(t) = |f(t)|$ is not differentiable.

What I find weird is that part (4) of the problem seems quite unrelated to the previous three parts of the question. The first three parts are specifically about the inner product as a multilinear function and the properties it and its derivative have, whereas for part (4) I can just take $f(x) = x$ and be done with it.

Typically, Spivak's problems are set up in a way to motivate some deeper idea. It is possible that he has slipped up here (after all, Calculus on Manifolds has numerous typos as well). But, I'd still like to ask,

Is there a way to look at part (4) of this problem in such a way that it connects to the inner product (or more precisely, to the previous three parts of the problem)?

real-analysis multivariable-calculus

share|cite|improve this question

asked 5 hours ago

Brahadeesh

3,19231143

add a comment | 

up vote
3
down vote

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up vote
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down vote

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Problem 2-13 from Spivak's Calculus on Manifolds asks the following:

Problem 2-13. Define $IP : mathbbR^n times mathbbR^n to mathbbR$ by $IP(x,y) = langle x, y rangle$.

1. Find $D(IP)(a,b)$ and $(IP)'(a,b)$.


2. If $f,g : mathbbR to mathbbR^n$ are differentiable and $h : mathbbR to mathbbR$ is defined by $h(t) = langle f(t), g(t) rangle$, show that $$h'(a) = langle f'(a)^T, g(a) rangle + langle f(a), g'(a)^T rangle.$$


3. If $f : mathbbR to mathbbR$ is differentiable and $|f(t)| = 1$ for all $t$, show that $langle f'(t)^T, f(t) rangle = 0$.


4. Exhibit a differentiable function $f : mathbbR to mathbbR$ such that the function $|f|$ defined by $|f|(t) = |f(t)|$ is not differentiable.

What I find weird is that part (4) of the problem seems quite unrelated to the previous three parts of the question. The first three parts are specifically about the inner product as a multilinear function and the properties it and its derivative have, whereas for part (4) I can just take $f(x) = x$ and be done with it.

Typically, Spivak's problems are set up in a way to motivate some deeper idea. It is possible that he has slipped up here (after all, Calculus on Manifolds has numerous typos as well). But, I'd still like to ask,

Is there a way to look at part (4) of this problem in such a way that it connects to the inner product (or more precisely, to the previous three parts of the problem)?

real-analysis multivariable-calculus

share|cite|improve this question

asked 5 hours ago

Brahadeesh

3,19231143

Problem 2-13 from Spivak's Calculus on Manifolds asks the following:

Problem 2-13. Define $IP : mathbbR^n times mathbbR^n to mathbbR$ by $IP(x,y) = langle x, y rangle$.

1. Find $D(IP)(a,b)$ and $(IP)'(a,b)$.


2. If $f,g : mathbbR to mathbbR^n$ are differentiable and $h : mathbbR to mathbbR$ is defined by $h(t) = langle f(t), g(t) rangle$, show that $$h'(a) = langle f'(a)^T, g(a) rangle + langle f(a), g'(a)^T rangle.$$


3. If $f : mathbbR to mathbbR$ is differentiable and $|f(t)| = 1$ for all $t$, show that $langle f'(t)^T, f(t) rangle = 0$.


4. Exhibit a differentiable function $f : mathbbR to mathbbR$ such that the function $|f|$ defined by $|f|(t) = |f(t)|$ is not differentiable.

What I find weird is that part (4) of the problem seems quite unrelated to the previous three parts of the question. The first three parts are specifically about the inner product as a multilinear function and the properties it and its derivative have, whereas for part (4) I can just take $f(x) = x$ and be done with it.

Typically, Spivak's problems are set up in a way to motivate some deeper idea. It is possible that he has slipped up here (after all, Calculus on Manifolds has numerous typos as well). But, I'd still like to ask,

Is there a way to look at part (4) of this problem in such a way that it connects to the inner product (or more precisely, to the previous three parts of the problem)?

real-analysis multivariable-calculus

share|cite|improve this question

asked 5 hours ago

Brahadeesh

3,19231143

share|cite|improve this question

share|cite|improve this question

asked 5 hours ago

Brahadeesh

3,19231143

asked 5 hours ago

Brahadeesh

3,19231143

asked 5 hours ago

Brahadeesh

3,19231143

3,19231143

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