If U^i is a contravariant vector and Vj is a covariant...

If $U^{i}$ is a contravariant vector and $V_{j}$ is a covariant vector, show that $U^{i} V_{j}$ is a $2^{\text {nd }}$ -rank mixed tensor. Hint: Write the transformation equations for $\mathbf{U}$ and $\mathbf{V}$ and multiply them.

Key Concepts

Contravariant Vectors

A contravariant vector is an object that transforms like the coordinates themselves under a change of basis. Its components change by the partial derivatives of the new coordinates with respect to the old coordinates, ensuring that the geometric or physical quantity it represents is independent of the coordinate system used.

Covariant Vectors

A covariant vector, also known as a dual vector or one-form, transforms in the opposite manner to contravariant vectors. Its transformation involves the partial derivatives of the old coordinates with respect to the new ones, making it the natural partner to contravariant vectors in forming scalar products and ensuring coordinate invariance.

Tensor Transformation Law

The tensor transformation law is a rule that describes how the components of a tensor change under a change of coordinates. For each index, the tensor components are multiplied by the appropriate factors (partial derivatives of the new coordinates with respect to the old or vice versa), and the covariance or contravariance determines which transformation matrix or its inverse is used, maintaining the tensor's intrinsic geometric meaning.

Mixed Tensors

A mixed tensor is an object that has both contravariant (upper) and covariant (lower) indices. Its transformation behavior involves a combination of the transformation laws for contravariant and covariant vectors. This mixed transformation property allows it to link different types of geometric quantities and ensures that the object remains invariant under coordinate transformations in the appropriate context.

Tensor Product

The tensor product of a contravariant vector and a covariant vector produces a second-rank tensor that is mixed in nature. By taking the product of their individual components and applying their transformation laws, one shows that the resulting object transforms according to the tensor transformation law, verifying its status as a tensor of mixed type.

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