 We can do little with tensors alone. We need to be able to do math with them. There are several operations with which we can combine two tensors A and B to form a new tensor C. We can add two tensors A and B of the same order, A i, j plus B i, j is equal C i, j. The result is a new tensor of the same order. If we represent the tensors A and B as matrices, then adding tensors is adding matrices component by component. The component A11 of the matrix A in the first row and first column is added to the component B11 of the matrix B, which is also in the same column and same row. The result is a 11 plus B11 in the first column and first row of the resulting matrix. This is how matrix addition works. We proceed in the same way with all other components. The result is the matrix C. We can subtract two tensors A and B of the same order, A i, j minus B i, j equals C i, j. The result is a new tensor C i, j of the same order. Subtraction works in the same way as addition. The next operation is probably new to you, namely the tensor product. Sometimes it is also called the outer product. Here, the components with equal indices are not combined as in the addition and subtraction of tensors. For the tensor product, we therefore have to label the indices of the tensors differently. The result of the tensor product A i, j with B k, m is a tensor C i, j, k, m. If we form the tensor product of tensors of the second order, the result is a tensor of the fourth order. If on the other hand, we form the tensor product of first order tensors A i and B k, the result is a second order tensor C i, k. This is how the tensor product works with two tensors of any order. The only exceptions are zero level tensors. In this case, the result remains a zero level tensor. The tensor sign is usually omitted in index notation. Let's take a concrete example of the tensor product that we can illustrate, namely the tensor product of first level tensors A and B. The two tensors are represented by the vectors with three components. The result is a second order tensor. This tensor represents a matrix. By definition, the first index, the index i, numbers the rows of the matrix, and the second index, the index k, numbers the columns. The tensor product does not necessarily have to be between two tensors of the same order. For example, we can also form the tensor product with the third order tensor A i, j, m, and the second level tensor B k, n. The result is a fifth order tensor C i, j, m, k, n. As you have probably already noticed, B k, n represents the k, n component of the tensor B, and A i, j, m is the i, j, m component of tensor A. If we form the tensor product, then this is the tensor product of the components. The result is the i, j, m, k, n component of the tensor C. When we write down a tensor with indices, we always mean its components. Nevertheless, we casually refer to its index notation as a tensor. Of course, the tensor product also works with superscript indices, which we will get to know in the next videos. If the indices i and j are at the top of the tensor A, then they must also be at the top of the resulting tensor C. The next operation we can perform is the contraction of a tensor. As an example, let's consider the fourth order tensor, T i, j, m, k. The contraction of this tensor means the following. We choose two of its indices. For example, the index i and m. Then we set the two indices equal. We can call them both i, for example. We then sum over the index i. In three-dimensional space, the index i goes from one to three, so the contraction of the tensor T gives the following sum. If we want to communicate these three steps to another physicist, then we say, contraction of the indices i and m of the tensor T. Or, contraction of the first and third indices of the tensor T. Contraction is very useful because it reduces the order of a tensor. For example, the contraction of the fourth level tensor T has reduced its order by two. The result of the contraction is a second order tensor, which we can call C j, k, for example. In physics, we use the Einstein summation convention, which states that we can omit the sum sign to simplify the notation if two identical indices appear in a tensor. In the tensor T i, j, i, k, in combination with the Einstein summation convention, the summation is performed using the double index i. If we contract a second order tensor T i, i, then the contraction is also called the trace of the tensor. The result is a zero order tensor that is a scalar. Of course, we can also contract the indices of two different tensors. For example, let's take a tensor m, i, j, and a tensor v, k. The tensor product m, i, j, v, k, without contraction, results in a third order tensor. Now we contract the indices j and k. In the matrix and vector representation, this corresponds to the multiplication of a matrix m with a vector v. The result ui is a first order tensor that is a vector.