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.

Contravariant vectors are essential objects in tensor analysis. They are vectors that transform in a specific way under a change of coordinates often represented as transformations using the inverse of Jacobian matrices.
When a coordinate system changes from \(x\) to \(x'\), a contravariant vector \(U^i\) transforms according to the following rule: \[U'^i = \frac{\text{d}x'^i}{\text{d}x^j} U^j \] Here, \(\frac{\text{d}x'^i}{\text{d}x^j}\) signifies the elements of the inverse Jacobian matrix that handles the coordinate transformation.
It's essential to note that contravariant vectors are often represented with upper indices. These upper indices indicate their nature and the way they transform.

Covariant vectors are another cornerstone in the realm of tensors, differing from contravariant vectors in their transformation behavior. Covariant vectors transform with the Jacobian matrix instead of its inverse.
For a coordinate transformation from \(x\) to \(x'\), a covariant vector \(V_j\) transforms as: \[V'_j = \frac{\text{d}x^i}{\text{d}x'^j} V_i \] In this case, \(\frac{\text{d}x^i}{\text{d}x'^j}\) represents the elements of the Jacobian matrix, directing how the vector components adjust to the new coordinates.
Covariant vectors usually have lower indices, and this index positioning signifies their transformation characteristics. Their transformation law helps in various operations and ensuring that the geometric and physical meanings are preserved across different coordinate systems.
Understanding both contravariant and covariant vectors is crucial as they combine to form more complex structures like mixed tensors.

A mixed tensor is an advanced object in tensor calculus that combines both contravariant and covariant components. Specifically, a 2nd-rank mixed tensor has one contravariant index and one covariant index.
In this context, consider the product of a contravariant vector \(U^i\) and a covariant vector \(V_j\). This product results in a mixed tensor: \[T^i{}_j = U^i V_j \] To understand why this is a mixed tensor, we examine how it transforms under a coordinate change. Through previously discussed transformations: \[U'^i V'_j = \left( \frac{\text{d}x'^i}{\text{d}x^k} U^k \right) \left( \frac{\text{d}x^l}{\text{d}x'^j} V_l \right) \] This simplifies to: \[T'^i{}_j = \frac{\text{d}x'^i}{\text{d}x^k} \frac{\text{d}x^l}{\text{d}x'^j} T^k{}_l \] The tensor \(T'^i{}_j\) follows the combined transformation laws for both contravariant and covariant components, making it a true mixed tensor.
Understanding mixed tensors is useful for more complex mathematical and physical systems.

Transformation equations are fundamental in tensor calculus. They show how tensor quantities change under coordinate transformations, preserving the tensor's structure across different frames of reference.
To illustrate this, consider a contravariant vector \(U^i\) and a covariant vector \(V_j\). Their respective transformation formulas are:

\[U'^i = \frac{\text{d}x'^i}{\text{d}x^k} U^k\] \[V'_j = \frac{\text{d}x^l}{\text{d}x'^j} V_l\]

When these vectors are combined to form a mixed tensor \(T^i{}_j = U^i V_j\), the transformation equation for \(T^i{}_j\) results from multiplying the individual transformation laws: \[T'^i{}_j = \left( \frac{\text{d}x'^i}{\text{d}x^k} U^k \right) \left( \frac{\text{d}x^l}{\text{d}x'^j} V_l \right) = \frac{\text{d}x'^i}{\text{d}x^k} \frac{\text{d}x^l}{\text{d}x'^j} T^k{}_l\] This shows that the transformation preserves the mixed nature of the tensor, ensuring that it properly adapts to the new coordinate system.
Mastery of transformation equations is key to working effectively with tensors, as it guarantees their correct interpretation in various coordinate frames.