10.2.5 Tensor Coordinate Change Area

Tensor Coordinate Change Area explains how tensors transform under coordinate changes, key to understanding invariance and geometric relationships.

Tensor Coordinate Change Area is the practical study of carrying out an actual coordinate transformation on tensor components, covering how to compute the required Jacobian matrix from explicit coordinate formulas and how to apply it to convert components between named coordinate systems such as Cartesian and polar.

Computing the Jacobian From Coordinate Formulas

Writing New Coordinates as Functions of Old Ones

The first practical step is expressing each new coordinate explicitly as a function of the old coordinates, which is the starting data needed before any Jacobian can be computed.

r = \sqrt{x^{2} + y^{2}}, θ = \arctan (\frac{y}{x})

Differentiating to Build the Matrix Entries

Each entry of the Jacobian matrix is obtained by differentiating one new coordinate with respect to one old coordinate, arranged systematically into a matrix.

\frac{\partial r}{\partial x} = \frac{x}{r}, \frac{\partial r}{\partial y} = \frac{y}{r}

Applying the Jacobian to Tensor Components

Transforming a Vector's Components

Once the Jacobian is assembled, transforming a contravariant vector's components between the two coordinate systems is a direct matrix multiplication, exactly as with a constant basis change matrix, but evaluated at the specific point in question.

v^{r} = \frac{\partial r}{\partial x} v^{x} + \frac{\partial r}{\partial y} v^{y}

Evaluating at a Specific Point

Because the Jacobian entries generally depend on position, the practical procedure requires evaluating each partial derivative at the specific point where the tensor's components are being transformed, rather than treating the matrix as a single fixed set of numbers valid everywhere.

Worked Pattern: Cartesian to Polar

Setting Up the Two Coordinate Systems

A standard worked pattern in this area transforms a vector's components from Cartesian coordinates ((x, y)) to polar coordinates ((r, \theta)), first writing the polar coordinates in terms of Cartesian ones, then computing all four Jacobian entries.

\frac{\partial θ}{\partial x} = - \frac{y}{r^{2}}, \frac{\partial θ}{\partial y} = \frac{x}{r^{2}}

Assembling and Applying the Full Matrix

Collecting all four partial derivatives into a single two-by-two matrix and multiplying it by the Cartesian component tuple produces the polar components directly, completing the coordinate change for that specific point.

Practical Cautions Within This Area

Recomputing at Every New Point

Because the Jacobian is position-dependent, a transformation valid at one point must be recomputed, using the same formulas but new numerical values, for every other point of interest; reusing a single numerical Jacobian across multiple points is a common practical mistake this area specifically warns against.

Avoiding Coordinate Singularities

Points where the Jacobian becomes singular, such as the origin in polar coordinates, must be excluded from this procedure entirely, since no valid transformation matrix exists there.

Visual Illustration

Why This Practical Area Is Necessary

Bridging the abstract coordinate transformation law with concrete coordinate formulas is what allows tensor components to be computed numerically in real curvilinear systems such as polar, cylindrical, or spherical coordinates. This area supplies the specific differentiation and evaluation steps that turn the general Jacobian-based transformation rule into a procedure that can be carried out point by point on an actual problem.