# Tangent Space to Product Manifold

Let $M$ and $N$ be smooth manifolds, and $p$ and $q$ be points on $M$ and $N$ respectively.

I want to show that $f:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN$ defined as $$f(Z)=(d\pi_M(Z),d\pi_N(Z))$$ is a linear isomorphism.

(I am using the derivations approach to tangent space).

To establish the isomorphism, it suffices to show that $f(Z)=0$ implies $Z=0$.

So let $f(Z)=0$ for some $Z\in T_{(p,q)}(M\times N)$.

Thus, by definition, it follows that $Z(\xi\circ \pi_M)=0$ and $Z(\zeta\circ \pi_N)=0$ for all $\xi\in \mathcal C^{\infty}(M)$ and $\zeta\in \mathcal C^{\infty}(N)$.

From here I need to show that $Z(\theta)=0$ for all $\theta \in \mathcal C^{\infty}(M\times N)$.

Can somebody see what to do to show the above?

Hint: $\theta = \theta(m,n)$ for $m \in M, n \in N$. How can you express the action of a differential operator $Z$ on this function?