Jacobians meaning geometrically When tackling double and triple integrals, we always use the Jacobian matrix when changing variables, such as to polar coordinates.
What actually is the Jacobian and what does it represent or mean geometrically? Also what does it represent geometrically in terms of infinitesimal areas $$ \delta x \delta y $$.
 A: The Jacobian matrix of a differentiable function $f\colon \mathbb{R}^m \to \mathbb{R}^n$ at a point $p \in \mathbb{R}^m$ is the matrix representing the linear transformation which is the best linear approximation to $f(x) - f(p)$ for $x \in \mathbb{R}^m$ near $p$. More precisely, if $J_p$ is the Jacobian matrix of $f$ at $p$, then we have
$$f(x) = f(p) + J_p(x - p) + o(x - p),$$
where the last term is little-o notation and means a function that approaches the zero vector (note that both sides of the above equation take values in $\mathbb{R}^n$) faster than linearly as $x \to p$. (This is a higher-dimensional analogue of the constant and linear terms of the Taylor series.) Stated without asymptotic notation,
$$\lim_{x \to p} \frac{\lVert f(x) - f(p) - J_p(x - p) \rVert}{\lVert x - p \rVert} = 0,$$
where $\lVert v \rVert$ is the Euclidean distance between a vector $v$ and the zero vector.
Geometrically, the graph of $f$ is an $m$-dimensional subset of $\mathbb{R}^m \times \mathbb{R}^n$, and the graph of $J_p$ is the tangent space to the graph of $f$ at the point $(p, f(p)) \in \mathbb{R}^m \times \mathbb{R}^n$.
The reason why the determinant of the Jacobian shows up when you do a change of coordinates is that the absolute value of the determinant of a linear transformation is the factor by which the transformation multiplies volumes. (For example, if you scale by a factor of $2$ in every direction in $\mathbb{R}^3$, then this multiplies volumes by a factor of $8$. If you apply a rigid motion of Euclidean space, it doesn't change volumes at all, and indeed, rigid motions have determinant $\pm 1$.) So the determinant of the Jacobian $J_p$ tells you the factor by which volumes are multiplied by $f$ locally at $p$ (which may vary with $p$).
