关于偏导数与雅可比行列式的符号表示困惑

Hey David, let's unpack this confusion around Jacobian determinants and their notation—this is a super common spot to get tripped up, so you're not alone! First, let's ground the explanation with your textbook examples:

I came across the following in a calculus textbook:
$$\frac{\partial(x,y)}{\partial(r,\theta)} = \left|\begin{matrix}\cos\theta & -r\sin\theta \ \sin\theta & r\cos\theta \end{matrix}\right| = r$$
$$\frac{\partial(y,z)}{\partial(r,\theta)} = \left|\begin{matrix}\sin\theta & r\cos\theta \ 1 & 0 \end{matrix}\right| = -r\cos\theta$$
$$\frac{\partial(x,z)}{\partial(r,\theta)} = \left|\begin{matrix}\cos\theta & -r\sin\theta \ 1 & 0 \end{matrix}\right| = r\sin\theta$$
These operations seem... I'm confused about the notation for partial derivatives and Jacobian determinants, and I hope to get an explanation.

First: What That Fraction-like Notation Actually Means #

The symbol $\frac{\partial(x_1, x_2, ..., x_n)}{\partial(u_1, u_2, ..., u_n)}$ is standard shorthand for the Jacobian determinant of a transformation from independent variables $(u_1,...,u_n)$ to dependent variables $(x_1,...,x_n)$.

Key rules to lock in:

• The numerator lists the variables you're expressing in terms of the denominator variables (dependent variables)
• The denominator lists the variables you're using to define the numerator variables (independent variables)
• This is not a regular fraction—you can't "cancel" variables or split it into separate partial derivatives. It's just a compact way to write the determinant of a partial derivative matrix.

Breaking Down Your First (Classic) Example #

Your first calculation is the standard Jacobian for converting polar coordinates to Cartesian coordinates ($x = r\cos\theta$, $y = r\sin\theta$). Here's how it comes together:

1. Build a matrix where each row corresponds to a dependent variable ($x$ then $y$), and each column corresponds to an independent variable ($r$ then $\theta$). Each entry is the partial derivative of the row variable with respect to the column variable:
   $$
   \begin{pmatrix}
   \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \
   \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}
   \end{pmatrix} = \begin{pmatrix}
   \cos\theta & -r\sin\theta \
   \sin\theta & r\cos\theta
   \end{pmatrix}
   $$
2. Take the determinant of this matrix:
   $$
   (\cos\theta)(r\cos\theta) - (-r\sin\theta)(\sin\theta) = r\cos^2\theta + r\sin^2\theta = r(\cos^2\theta + \sin^2\theta) = r
   $$
   This Jacobian tells us how area scales when converting from polar to Cartesian coordinates.

Your Other Examples: Subset Jacobians for Surface Integrals #

The second and third examples look like they're designed for computing surface integrals (where we project a 3D surface onto a 2D plane). Let's assume we're working with a surface where $z = r$ (that explains the $\frac{\partial z}{\partial r} = 1$ entry in the matrices).

For $\frac{\partial(y,z)}{\partial(r,\theta)}$:

• We build the matrix from partial derivatives of $y$ and $z$ with respect to $r$ and $\theta$:
  $$
  \begin{pmatrix}
  \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \
  \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta}
  \end{pmatrix} = \begin{pmatrix}
  \sin\theta & r\cos\theta \
  1 & 0
  \end{pmatrix}
  $$
• The determinant is calculated as $(\sin\theta)(0) - (r\cos\theta)(1) = -r\cos\theta$, which matches your textbook's result.

The same logic applies to $\frac{\partial(x,z)}{\partial(r,\theta)}$—swap $y$ with $x$ in the matrix, compute the determinant, and you get $r\sin\theta$.

Clearing Up Common Notation Mix-Ups #

To avoid confusion going forward:

• Partial derivatives (e.g., $\frac{\partial x}{\partial r}$) are single derivatives: they measure how one dependent variable changes when one independent variable is varied (others held constant).
• Jacobian determinants (e.g., $\frac{\partial(x,y)}{\partial(r,\theta)}$) are determinants of partial derivative matrices: they describe how area/volume/orientation changes under a transformation.
• The sign of the Jacobian matters for orientation (e.g., whether a transformation flips the direction of vectors), which is why your textbook keeps the negative sign in the second example—critical for surface integrals where orientation affects the final result.

内容的提问来源于stack exchange，提问作者David Parks