Is the product of a differential always equal to the differential of the product?

Asked Feb 07 '21 at 06:20

Active Feb 07 '21 at 06:20

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Let $f \times g:X \times Y \rightarrow Z \times W$ be the product of two smooth functions $f:X \rightarrow Z$ and $g:Y \rightarrow W$ in the category of smooth manifolds.

My question is the following:

Is $(f \times g)_{*,{(p,q)}}= f_{*,p} \times g_{*,q}$ holds true for all $(p,q) \in X \times Y$? (Here "*" denotes the differential)

My guess is yes , because of the fact that $T_{(p,q)}(X \times Y)$ is isomorphic to $T_{p}X \times T_qY$.

Is my guess correct?

(Thanks in advance)

asked Feb 07 '21 at 06:20

mathematics student

1

Can you prove it for euclidean spaces? – EternalBlood Feb 07 '21 at 06:27
@EternalBlood Yes, I guess. – mathematics student Feb 07 '21 at 06:30
1

It is actually easier. What do you get if you write such isomorphism explicitely? Just evaluate both differentials and write the isomorphism between the two tangent spaces, and see what happens. You can look at the isomorphism here if you don't know it already https://math.stackexchange.com/questions/413766/tangent-space-of-product-manifold – EternalBlood Feb 07 '21 at 06:47
@EternalBlood Thanks. So , my guess is correct right? – mathematics student Feb 07 '21 at 06:50
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Yes, but you should try to convince yourself once and for all. It is very easy if you do what I said above, or you might try to do it just for euclidean spaces and use charts. – EternalBlood Feb 07 '21 at 06:55
Thanks , I will definitely do. – mathematics student Feb 07 '21 at 06:56

Is the product of a differential always equal to the differential of the product?

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