Tensor calculus

Tensors notation allows a vector (${\displaystyle {\vec {V}}}$) to be decomposed into an Einstein summation representing the tensor contraction of a basis vector (${\displaystyle {\vec {Z}}_{i}}$ or ${\displaystyle {\vec {Z}}^{i}}$) with a component vector (${\displaystyle V_{i}}$ or ${\displaystyle V^{i}}$).

${\displaystyle {\vec {V}}=V^{i}{\vec {Z}}_{i}=V_{i}{\vec {Z}}^{i}}$

Every vector has two different representations, one referred to as contravariant component (${\displaystyle V^{i}}$) with a covariant basis (${\displaystyle {\vec {Z}}_{i}}$), and the other as a covariant component (${\displaystyle V_{i}}$) with a contravariant basis (${\displaystyle {\vec {Z}}^{i}}$). Tensor objects with all upper indexes are referred to as contravariant, and tensor objects with all lower indexes are referred to as covariant. The need to distinguish between contravariant and covariant arises from the fact that when we dot an arbitrary vector with its basis vector related to a particular coordinate system, there are two ways of interpreting this dot product, either we view it as the projection of the basis vector onto the arbitrary vector, or we view it as the projection of the arbitrary vector onto the basis vector, both views of the dot product are entirely equivalent, but have different component elements and different basis vectors:

$${\displaystyle {\vec {V}}\cdot {\vec {Z}}_{i}=V_{i}={\vec {V}}^{T}{\vec {Z}}_{i}={\vec {Z}}_{i}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}^{i}}({\vec {V}})}\cdot {\vec {Z}}_{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}^{i})}\cdot {\vec {V}}}$$

$${\displaystyle {\vec {V}}\cdot {\vec {Z}}^{i}=V^{i}={\vec {V}}^{T}{\vec {Z}}^{i}={{\vec {Z}}^{i}}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}_{i}}({\vec {V}})}\cdot {\vec {Z}}^{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}_{i})}\cdot {\vec {V}}}$$

For example, in physics you start with a vector field, you decompose it with respect to the covariant basis, and that's how you get the contravariant coordinates. For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical there is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.

${\displaystyle {\vec {V}}=V^{i}{\vec {Z}}_{i}}$

variable	description	Type
${\displaystyle {\vec {V}}}$	vector	invariant
${\displaystyle V^{i}}$	contravariant components (ordered set of scalars)	variant
${\displaystyle {\vec {Z}}_{i}}$	covariant bases (ordered set of vectors)	variant

${\displaystyle {\vec {V}}=V_{i}{\vec {Z}}^{i}}$

variable	description	type
${\displaystyle {\vec {V}}}$	vector	invariant
${\displaystyle V_{i}}$	covariant components (ordered set of scalars)	variant
${\displaystyle {\vec {Z}}^{i}}$	contravariant bases (ordered set of covectors)	variant

The metric tensor represents a matrix with scalar elements (${\displaystyle Z_{ij}}$ or ${\displaystyle Z^{ij}}$) and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa.

Example of lowering index using metric tensor:

${\displaystyle T_{i}=Z_{ij}T^{j}}$

Example of raising index using metric tensor:

${\displaystyle T^{i}=Z^{ij}T_{j}}$

The metric tensor is defined as:

${\displaystyle Z_{ij}={\vec {Z}}_{i}\cdot {\vec {Z}}_{j}}$

${\displaystyle Z^{ij}={\vec {Z}}^{i}\cdot {\vec {Z}}^{j}}$

This means that if we take every permutation of a basis vector set and dotted them against each other, and then arrange them into a square matrix, we would have a metric tensor. The caveat here is which of the two vectors in the permutation is used for projection against the other vector, that is the distinguishing property of the covariant metric tensor in comparison with the contravariant metric tensor.

Two flavors of metric tensors exist: (1) the contravariant metric tensor (${\displaystyle Z^{ij}}$), and (2) the covariant metric tensor (${\displaystyle Z_{ij}}$). These two flavors of metric tensor are related by the identity:

${\displaystyle Z_{ik}Z^{jk}=\delta _{i}^{j}}$

For an orthonormal Cartesian coordinate system, the metric tensor is just the kronecker delta ${\displaystyle \delta _{ij}}$ or ${\displaystyle \delta ^{ij}}$, which is just a tensor equivalent of the identity matrix, and ${\displaystyle \delta _{ij}=\delta ^{ij}=\delta _{j}^{i}}$.

In addition a tensor can be readily converted from an unbarred(${\displaystyle x}$) to a barred coordinate(${\displaystyle {\bar {x}}}$) system having different sets of basis vectors:

$${\displaystyle f(x^{1},x^{2},\dots ,x^{n})=f{\bigg (}x^{1}({\bar {x}}),x^{2}({\bar {x}}),\dots ,x^{n}({\bar {x}}){\bigg )}={\bar {f}}({\bar {x}}^{1},{\bar {x}}^{2},\dots ,{\bar {x}}^{n})={\bar {f}}{\bigg (}{\bar {x}}^{1}(x),{\bar {x}}^{2}(x),\dots ,{\bar {x}}^{n}(x){\bigg )}}$$

by use of Jacobian matrix relationships between the barred and unbarred coordinate system (${\displaystyle {\bar {J}}=J^{-1}}$). The Jacobian between the barred and unbarred system is instrumental in defining the covariant and contravariant basis vectors, in that in order for these vectors to exist they need to satisfy the following relationship relative to the barred and unbarred system:

Contravariant vectors are required to obey the laws:

${\displaystyle v^{i}={\bar {v}}^{r}{\frac {\partial x^{i}({\bar {x}})}{\partial {\bar {x}}^{r}}}}$

${\displaystyle {\bar {v}}^{i}=v^{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}}$

Covariant vectors are required to obey the laws:

${\displaystyle v_{i}={\bar {v}}_{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}}$

${\displaystyle {\bar {v}}_{i}=v_{r}{\frac {\partial x^{r}({\bar {x}})}{\partial {\bar {x}}^{i}}}}$

There are two flavors of Jacobian matrix:

1. The J matrix representing the change from unbarred to barred coordinates. To find J, we take the "barred gradient", i.e. partial derive with respect to ${\displaystyle {\bar {x}}^{i}}$:

${\displaystyle J={\bar {\nabla }}f(x({\bar {x}}))}$

2. The ${\displaystyle {\bar {J}}}$ matrix, representing the change from barred to unbarred coordinates. To find ${\displaystyle {\bar {J}}}$, we take the "unbarred gradient", i.e. partial derive with respect to ${\displaystyle x^{i}}$:

${\displaystyle {\bar {J}}=\nabla {\bar {f}}({\bar {x}}(x))}$

Tensor calculus provides a generalization to the gradient vector formula from standard calculus that works in all coordinate systems:

${\displaystyle \nabla F=\nabla _{i}F{\vec {Z}}^{i}}$

Where:

${\displaystyle \nabla _{i}F={\frac {\partial F}{\partial Z^{i}}}}$

In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. the polar gradient vector formula vs. the spherical gradient vector formula, etc.). In standard calculus, each coordinate system has its own specific formula, unlike tensor calculus that has only one gradient formula that is equivalent for all coordinate systems. This is made possible by an understanding of the metric tensor that tensor calculus makes use of.