The following is an illustration of the Jacobian in my text, along with part of the explanation (also see similar explanation here):
Since the side-lengths are small, by the Mean Value Theorem, we have
$$x(u + ∆u, v) − x(u, v) ≈
{∂x\over {∂u}}(α_1,v)∆u,\\\ \text{where} \\u ≤ α_1 ≤ u + ∆u$$
I am trying to understand this, specifically four things (I am not sure if they are related):
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What does the Mean Value Theorem have to do with this, and what does it have to do with the length being small?
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What is the purpose of this new variable, $a_1$, that is defined as being between $u$ and $u + ∆u$?
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What is the purpose of the partial derivative?
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What is purpose of the $∆u$ that is multiplied by the partial derivative?
Best Answer
Remember we can think of matrices as representing linear maps from one space to another. The Jacobian is a matrix which represents the map from $U \times V \to X \times Y$ and approximates this distortion linearly. This means that $(x,y)=(x(u,v), y(u,v))$ might not be linear. It might twist and warp points in strange ways, but we can approximate that with the Jacobian. Explicitly, we can write the Jacobian as
$$J = \begin{bmatrix} \dfrac{\partial x(u,v)}{\partial u} & \dfrac{\partial x(u,v)}{\partial v}\\ \dfrac{\partial y(u,v)}{\partial u} & \dfrac{\partial y(u,v)}{\partial v} \end{bmatrix}$$
When we say approximation we mean we can find approximately where the new point maps to by $$\begin{bmatrix} x(u + \Delta u, v +\Delta v)\\ y(u+\Delta u, v+\Delta v) \end{bmatrix} \approx \begin{bmatrix} x(u,v)\\ y(u,v) \end{bmatrix} + \begin{bmatrix} \dfrac{\partial x(u,v)}{\partial u} & \dfrac{\partial x(u,v)}{\partial v}\\ \dfrac{\partial y(u,v)}{\partial u} & \dfrac{\partial y(u,v)}{\partial v} \end{bmatrix} \begin{bmatrix} \Delta u\\ \Delta v \end{bmatrix}$$
We can see how this works in one dimension (holding $v$ constant) by looking at the definition of (partial) derivative $$\dfrac{\partial x(u,v)}{\partial u} = \lim_{h \to 0} \dfrac{x(u+h, v) - x(u,v)}{h}$$
When $\Delta u$ is sufficiently small then we can approximate as $$\dfrac{\partial x(u,v)}{\partial u} \Delta u \approx x(u+\Delta u, v) - x(u,v)$$
Earlier we mentioned twisting and warping. One thing we might be interested in is how the length of a line segment changes. Say I know the length of the bottom of the rectangle, which is given by $(u+\Delta u) - u$. I want to know how long $(x(u,v), x(u + \Delta u, v))$ is.
By the mean value theorem, there is some point $\alpha_1 \in (u, u + \Delta u)$ such that
$$\dfrac{\partial x(u,v)}{\partial u} (\alpha_1, v) = \dfrac{x(u + \Delta u, v) - x(u,v)}{\Delta u}$$
Thus the length given by $x(u + \Delta u, v) - x(u,v)$ can be found with
$$\dfrac{\partial x(u,v)}{\partial u} (\alpha_1, v) \Delta u=x(u + \Delta u, v) - x(u,v)$$
I agree I'm a bit confused by the approximation if we're invoking mean value theorem, which will give exactness.