 Before we understand tensors in their general definition, let's first get to know them from an engineering perspective. As long as you know scalars, vectors, and matrices, it will be easy for you to understand tensors from this perspective because tensors are nothing more than a generalization of scalars, vectors, and matrices. Just as we use scalars, vectors, and matrices to describe physical laws, we can use tensors to describe physics. Tensors are an even more powerful tool with which we can describe physics that cannot be formulated with scalars, vectors, and matrices alone. In order to develop the modern theory of gravity, Albert Einstein first had to understand the concept of tensors. Only then was he able to mathematically formulate the general theory of relativity. Let's start with the simplest tensor, the zero-order tensor. Represented by a scalar, let's call it sigma. This tensor is therefore an ordinary number and represents, for example, the electric conductivity of an isotropic wire. The zero-th order tensor as electrical conductivity therefore indicates how well a wire conducts electric current. A somewhat more complex tensor, let's call it J, is a first-order tensor. This is a vector with three components, J1, J2, and J3 in three-dimensional space. Of course, the vector can also describe two components or more than three components as is the case in the general theory of relativity where we work with tensors in a four-dimensional space time. This first-order tensor can, for example, describe the current density in a wire. We have represented the first-order tensor as a column vector. Of course, we can also represent it as a row vector. At this stage, it doesn't matter how we write the components, but remember that it will matter later. The notation of first-order tensors as row or column vectors only makes sense when we are calculating with concrete numbers such as in computer physics, where we use tensors to obtain numerical results. In order to work with them theoretically, for example, to derive equations or simply to formulate a physical theory, the tensors are formulated compactly in index notation. You are probably already familiar with this concept from vector calculus. Instead of writing out all three components of the first-order tensor, we write them with an index k. How we label the index is irrelevant. jk stands for the first component j1, second component j2, or third component j3, depending on what we insert for index k. In theoretical physics, we usually do not insert anything specific because we want to write the physics as generally and compactly as possible. It is not clear from this index notation jk whether it represents a column or row vector. This is not good because later it will be important to distinguish between column and row vectors. But we can easily introduce this distinction into our index notation by noting the index below if we mean a row vector, and we notate the index above if we mean a column vector. The notation of indices above and below has a deeper meaning, which we will get to know later. At this stage, we are only distinguishing the representation of the first-order tensor. The next most complex tensor is the second-order tensor. Let's call this tensor a sigma because a second-order tensor can also describe the electrical conductivity like a zero-level tensor. Conductivity as a zero-order tensor describes an isotropic material. The conductivity as a second-order tensor, on the other hand, describes a non-isotropic material in which the conductivity is different depending on the direction in which the current flows. You have probably already become familiar with this tensor in mathematics and the matrix representation. In a three-dimensional space, the second-order tensor is a three-by-three matrix. We also use index notation for the second-order tensor and write the components of the matrix as sigma mk, for example. The indices m and k can have the values 1, 2, or 3. The index m indicates the row and the index k indicates the column. We can continue the game and consider a third-order tensor. This tensor has three indices, sigma mk, n. The fourth-order tensor has four indices, sigma m, k, n, i. The indices of a tensor of any level can also be superscripted. For example, the indices mk of the fourth-level tensor can be at the top and the indices ni at the bottom. You will learn what this means in the next videos.