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Let $f:\mathbb{R}^2\to \mathbb{R}^2$ be a differentiable map. If I write elements of $\mathbb{R}^2$ as $(x,y)$ and the components of $f$ as $f=(f_1,f_2)$ then its Jacobian Matrix is given as:

$J= \left(\begin{array}{cc} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{array}\right)$.

However, I can consider $f$ as a continous map between the 2-dimensional real manifolds $\mathbb{R}^2$. Its differential is then given as

$$df|p:T_p\mathbb{R}^2\to T_p\mathbb{R}^2$$ where a tangent vector $[\gamma]$ is mapped to the tangent vector $[f\circ\gamma]$.

As $T_p\mathbb{R}^2\cong \mathbb{R}^2$ this differential map can be described by an 2x2-matrix, which should be exactly the jacobian Matrix $J$ as above.

My Question: I wonder how one can that the matrix representation of $df|p$ actually IS the Jacobian of $f$ and would be thankful for an explicit calculation of this.

Thanks!

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  • $\begingroup$ The Jacobian is a sort of separate entity multiplier between $ f1,f2 $ deferentially connected with common independent parameters $ x,y $ or variables. $\endgroup$
    – Narasimham
    Feb 28 '16 at 18:28
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Actually, this is mostly basic linear algebra. To get a matrix representation $A$ of a linear map $L: V\rightarrow W$, you have to choose basises on $V$ and $W$. Let $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_m\}$ be those sets, respectively. Then the components of $A$ are given by the expansion $$ Lv_i = \sum_j A_{ji} w_j. $$ In your case $V=W=T_pM$ and $L=df_p$, the basis is given by the coordinate vector fields of the choosen chart, i.e. the identity. By definition, for every smooth function $h: \mathbb{R}^2\rightarrow \mathbb{R}$ it holds: $$ df_p(\partial_{i})(h)=\partial_i(h\circ f)|_p. $$ In particular, if we choose the projection on the $j$-th component $h=pr_j$ (which is the $j$-th component of our chart): $$ df_p(\partial_{i})(pr_j)=\partial_i(f_j)|_p. $$ On the other hand, we have the above expansion: $$ df_p(\partial_{i})(pr_j)=\sum_k J_{ki} \partial_k(pr_j)|_p = \sum_k J_{ki} \delta_{kj} = J_{ji}. $$ Hence, $$ J_{ij}=\partial_j(f_i)|_p = \frac{\partial f_i}{\partial x_j}(p). $$

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