🧪The Metric Tensor Covariant Derivatives and Tensor Densities

A tensor field assigns a tensor to each point in space, with the metric tensor being a crucial example that defines distance and inner products, whose derivatives are determined by Christoffel symbols; these components are essential for defining the covariant derivative, which extends differential operators like the gradient, divergence, and curl to general coordinate systems, while a tensor density is a generalization of a tensor that includes an extra scaling factor related to the Jacobian of coordinate transformations.

🎬how the metric tensor changes with the geometry of a coordinate system

The 3D visualization demonstrates that the metric tensor acts as a dynamic blueprint for a coordinate system's geometry. While the diagonal components of the tensor stay constant at one (since the basis vectors have unit length), the off-diagonal components become non-zero when the grid is skewed, with their values directly encoding the angles between the non-orthogonal axes. This shows how the tensor precisely captures the distortions of the space, allowing for calculations of distances and angles even in a non-Cartesian system.

🎬a non-orthogonal coordinate system dynamically calculating and displaying the metric tensor and its inverse

The metric tensor ( $g_{a b}$ ) is a matrix that defines distances in a coordinate system, with non-zero off-diagonal elements indicating non-perpendicular axes in a non-orthogonal system, like our skewed grid example, while its components can also vary with position, as seen with the $r^2$ term in polar coordinates, and its inverse $\left(g^{a b}\right)$ is crucial for raising/lowering indices but requires a more complex calculation than simple reciprocals, contrasting sharply with the identity matrix of a simple Cartesian system.

🎬how the metric tensor in polar coordinates is used to compute the circumference of a circle

The metric tensor, $g_{a b}$, defines how distances are measured in any coordinate system, especially non-Cartesian ones. A metric tensor with non-zero off-diagonal elements, as seen in the skewed grid example, signifies non-orthogonal axes, while a diagonal matrix with non-unity elements, like the $g_{\phi \phi}=\rho^2$ term in polar coordinates, reveals how coordinate components change in length with position. The inverse metric tensor, $g^{a b}$, is essential for calculations in such systems, underscoring that Cartesian coordinates are a simplified, special case where the metric tensor is just the identity matrix.

🎬compares a simple linear coordinate system with zero Christoffel symbols to a curvilinear system with non-zero

Christoffel symbols are a measure of how a coordinate system's basis vectors change from point to point. In the linear coordinate system on the left, the basis vectors (v₁ and v₂) are the same everywhere. Since there is no change, their derivatives are zero, which means all Christoffel symbols are zero. This holds true even though the system is non-orthogonal. In the spherical (curvilinear) coordinate system on the right, the basis vectors (∂/∂r and ∂/∂θ) constantly change direction. This change is precisely what the non-zero Christoffel symbols account for.

🫧Cue Column

The Metric Tensor Covariant Derivatives and Tensor Densities | Condensed Notesviadean on Notion