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.
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A tensor field associates a tensor to every point in a given space, extending the concepts of scalar and vector fields. Scalars and vectors are themselves specific types of tensors, making their fields the most natural examples of tensor fields. The components of a tensor field are functions of the base space, and the basis vectors themselves can also vary from point to point.
The metric tensor is a fundamental tensor field, typically of type ( 0,2 ), that defines the distance between nearby points and the inner product between tangent vectors in a space. It is a symmetric and positive-definite tensor. The inverse metric tensor (type ( 2,0 )) is its inverse in a matrix sense and is used to raise indices. The metric tensor also defines the line element, which is crucial for calculating the length of a curve.
The covariant derivative is a generalization of the partial derivative for tensor fields that accounts for the change in the basis vectors from one point to another. Unlike partial derivatives of components, covariant derivatives transform as tensors. Christoffel symbols are the components that describe how the basis vectors change with position. They are symmetric in their lower indices and can be expressed in terms of the metric tensor and its derivatives. The covariant derivative of the metric tensor is always zero.
The concepts of gradient, divergence, and curl can be generalized using the covariant derivative. The gradient of a tensor field, $\nabla T$, is a tensor with a rank one higher than the original. The divergence is the contraction of the gradient, reducing the rank by one. The Laplace operator is the divergence of the gradient. The curl of a vector field is related to the antisymmetric part of its covariant derivative, which is equivalent to the antisymmetric part of the partial derivatives due to the symmetry of the Christoffel symbols. This concept can be extended to higher-rank tensors by anti-symmetrizing their indices.
A tensor density is a quantity whose transformation rule is a normal tensor transformation multiplied by a power of the Jacobian determinant, $J^w$. This is useful for concepts like coordinate density, which changes depending on the coordinate system. Tensors are just a special case of tensor densities with a weight ( $w$ ) of zero. The permutation symbol can be viewed as a tensor density with a weight of +1 or -1, depending on whether it's contravariant or covariant.
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