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How do you raise the indices of a tensor?
We raise an index by aplying the metric to a tensor, like this Aμ=gμνAν. Now, if you want to raise two index you need to operate with the metric twice. In a more formal language lowering and raising indices is a way to construct isomorphisms between covariant and contravariant tensorial spaces.
What are upper and lower indices?
When a vector space is equipped with an inner product (or metric as it is often called in this context), there exist operations that convert a contravariant (upper) index into a covariant (lower) index and vice versa. This operation is called raising an index.
What is contravariant metric tensor?
The contravariant metric tensor just measures the dot products of these vectors, so we can have an idea of how to measure lengths with them. For instance, take u=y−a(x) as you gave us. Taking the gradient of u, we get. gu=∇u=(gx∂x+gy∂y+gz∂z)u(x,y,z)=gy−a′(x)gx.
How do you write a metric tensor?
In a local coordinate system xi, i = 1, 2, …, n, the metric tensor appears as a matrix, denoted here by G, whose entries are the components gij of the metric tensor relative to the coordinate vector fields. is called the first fundamental form associated to the metric, while ds is the line element.
How do you contract two tensors?
In tensor index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.
How do you raise the index of a metric tensor?
Multiplying by the contravariant metric tensor gij and contracting produces another tensor with an upper index: The same base symbol is typically used to denote this new tensor, and repositioning the index is typically understood in this context to refer to this new tensor, and is called raising the index, which would be written
How do you find the covariant lower indexed tensor?
To obtain the covariant tensor Fαβ, multiply by the metric tensor and contract: and by antisymmetry, for α = k = 1, 2, 3, β = 0 : The (covariant) lower indexed tensor is then:
What is the meaning of raising and lowering indices?
Raising and lowering indices From Wikipedia, the free encyclopedia In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Is it possible to contract a tensor?
A metric itself is a (symmetric) (0,2)-tensor, it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric. This produces a new tensor with the same index structure as the previous, but with lower index in the position of the contracted upper index.