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I am new in differential geometry and I stuck in a basic question about partial derivatives on a manifold.

Let $M$ be a manifold defined as

$$ M = \{(x,y,z)~|~ x^2+y^2 +z^2 = 1\}.$$

Now let the function $f(x,y,z) = x^2 +y^2+z^2$ defined on $M$ (it means that $f: M \rightarrow \mathbb{R}).$

My question is what $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$ means and what are their values?

Thanks for any explanation.

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    Your function is constant on $M$.. and $\frac{\partial f}{\partial x}$ does not make sense on $M$ (it does on $\mathbb R^3$) – 2017-01-09
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    Once you choose local coordinates $(x^1,x^2)$ on an open subset of $M$, then $\dfrac{\partial f}{\partial x^i}$ will make sense. – 2017-01-09

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The notion that you can choose here to compute the "partial derivative" at a point $u\in S^2$ is to consider the tangent vector space $T_uS^2$, $v\in T_uS^2$, consider $df_u(v)$.

There is the notion of connection to compute derivative of tensor fields.

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    There's no $\mathbb S^3$ in the question – 2017-01-09