Question

WHAT IS MEAN BY (0,2) and (2,0) (1,1) tensor.

Answer 1

Answer on Question #63035 – Math – Matrix | Tensor Analysis

Question

WHAT IS MEANT BY (0,2), (2,0) and (1,1) tensor?

Solution

The tensors are classified according to their type $(n,m)$, where $n$ is the number of contravariant indices, $m$ is the number of covariant indices, and $n + m$ gives the total order of the tensor.

A mixed tensor of rank or order $(m + n)$

$$T _ {j _ {1} j _ {2} \dots j _ {n}} ^ {i _ {1} i _ {2} \dots i _ {m}}$$

is contravariant of order $m$ and covariant of order $n$ if it obeys the transformation law

$$\widetilde {T} _ {j _ {1} j _ {2} \dots j _ {n}} ^ {i _ {1} i _ {2} \dots i _ {m}} = \left[ J \left(\frac {x}{\bar {x}}\right) \right] ^ {W} T _ {b _ {1} b _ {2} \dots b _ {n}} ^ {a _ {1} a _ {2} \dots a _ {m}} \frac {\partial \bar {x} ^ {i _ {1}}}{\partial x ^ {a _ {1}}} \frac {\partial \bar {x} ^ {i _ {2}}}{\partial x ^ {a _ {2}}} \dots \frac {\partial \bar {x} ^ {i _ {m}}}{\partial x ^ {a _ {m}}} \cdot \frac {\partial x ^ {b _ {1}}}{\partial \bar {x} ^ {j _ {1}}} \frac {\partial x ^ {b _ {2}}}{\partial \bar {x} ^ {j _ {2}}} \dots \frac {\partial x ^ {b _ {n}}}{\partial \bar {x} ^ {j _ {n}}},$$

where

$$J \left(\frac {x}{\bar {x}}\right) = \left| \frac {\partial x}{\partial \bar {x}} \right| = \frac {\partial (x ^ {1} , x ^ {2} , \ldots , x ^ {N})}{\partial (\bar {x} ^ {1} , \bar {x} ^ {2} , \ldots , \bar {x} ^ {N})}$$

is the Jacobian of the transformation.

When $W = 0$ the tensor is called an absolute tensor, otherwise it is called a relative tensor of weight $W$.

A tensor field of type (2,0) on the n-dimensional smooth manifold M associates with each $x$ a collection of $n^2$ smooth functions $T^{ij}(x^1,x^2,\ldots ,x^n)$ which satisfy the following transformation rule:

$$\widetilde {T} ^ {i j} = \frac {\partial \bar {x} ^ {i}}{\partial x ^ {k}} \frac {\partial \bar {x} ^ {j}}{\partial x ^ {m}} T ^ {k m} \quad (\text{'contravariant rank 2'})$$

Inverse metric tensor, bivector are examples of a (2,0)-tensor.

A tensor field of type (1,1) on the n-dimensional smooth manifold M associates with each $x$ a collection of $n^2$ smooth functions $F_{j}^{i}(x^{1},x^{2},\ldots ,x^{n})$ which satisfy the following transformation rule:

$$\widetilde {F} _ {j} ^ {i} = \frac {\partial \bar {x} ^ {i}}{\partial x ^ {k}} \frac {\partial x _ {m}}{\partial \bar {x} ^ {j}} F _ {m} ^ {k} \quad (\text{'mixed with contravariant rank 1 and covariant rank 1'})$$

A linear transformation is an example of a (1,1)-tensor.

A tensor field of type (0,2) on the n-dimensional smooth manifold M associates with each $x$ a collection of $n^2$ smooth functions $E_{ij}(x^1, x^2, \ldots, x^n)$ which satisfy the following transformation rule:

$$\widetilde{E}_{ij} = \frac{\partial x_k}{\partial \bar{x}^j} \frac{\partial x_m}{\partial \bar{x}^j} E_{km} \quad (\text{'covariant rank 2'})$$

A bilinear form is an example of a (0,2)-tensor.

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