4.1. Multiplication of a second order tensor with a vector

When a second order tensor is multiplied with a first order tensor, then the second vector from the tensor gets dotted with the first order tensor. This is how the multiplication is defined:

In this chunk, we learn how to multiply a second order tensor (which can be thought of as a matrix) with a first order tensor (a vector). The key operation here is the dot product, where components from the second vector of the tensor are combined with the first order tensor, resulting in a new vector. This means that while we perform the multiplication, we apply the rules of vector dot products, which helps us understand how one tensor interacts with another.

Imagine you have a tool kit (the tensor) and one specific tool (the vector). To use the tool effectively, it needs to fit into the toolkit properly. This multiplication is like taking that specific tool (vector) and seeing how it fits into the overall tool kit (the tensor), giving you a new way to use the tool.

Here, δ is the Kronecker delta function and is defined as: δ_{ij} = 1 if i = j, and 0 if i ≠ j

The Kronecker delta function δ is a simple yet powerful mathematical tool used in tensor multiplication. It serves as a way to isolate components in the matrix form of the tensors during multiplication. If the indices of the delta function match (i = j), it equals 1, denoting that we keep that term in our calculations. If they do not match (i ≠ j), it drops out, making our calculations simpler and more organized.

Think of the Kronecker delta as a bouncer at a club. If your name on the guest list (indices i and j) matches, you get in (the value is 1). If not, the bouncer denies you entry (the value is 0). It ensures only the 'matching entries' are considered in our multiplication process.

Now consider the summation over k in (18), due to the Kronecker delta function present there, only the terms having j = k will contribute to the summation and the others will be zero. Thus, we can get rid of the summation over k and replace k by j at all places.

This chunk explains how the presence of the Kronecker delta allows for simplifications in the multiplication process. Since the delta function ensures only the matching terms count, we can effectively replace the variable of summation k with j, avoiding unnecessary calculations. This simplification makes the math cleaner and more straightforward, which aids in understanding how tensor multiplication works.

Think of this as organizing files in a filing cabinet. If you have rules that only apply to certain files (like the Kronecker delta), you can quickly decide which files to look at (the relevant terms) and ignore the rest. This streamlining helps you work faster and more efficiently.

Thus, when we multiply a second order tensor with a vector, we get a vector whose components are given by...

In this chunk, we see the final result of multiplying a second order tensor with a vector. The outcome is a new vector formed from the components aligned according to the multiplication rules established earlier. This vector represents a transformed version of the original vector, influenced by the properties of the tensor it was multiplied with.

Imagine that you have a recipe (the tensor) that affects how you prepare a dish (the vector). After following the recipe, the resulting dish (the resulting vector) will taste different (or have new characteristics) due to the ingredients and methods specified in the recipe. The multiplication of the tensor with the vector similarly transforms the original vector into something new.

Thus, we simply multiply the matrix representation of C with the column form of b to get the column form of the resulting vector a.

This chunk highlights the visual aspect of tensor multiplication. We can represent the operation as a matrix multiplication. When we visualize the second order tensor as a matrix C and the vector as a column vector b, we can perform the multiplication operation in a straightforward manner that most students already have experience with from algebra. This helps reinforce the mechanical process behind tensor operations.

Consider using a blender to mix fruits (vector b) with yogurt (tensor C). When you turn on the blender (the multiplication operation), the ingredients combine to create a smoothie (resulting vector a). The blender here symbolizes the tensor, transforming the individual ingredients (vector) into a final product through the process of blending.

Let us recall the cross product definition in (7) where we had written it as a skew-symmetric matrix times a vector. On further noting the multiplication we just saw, we immediately conclude that the cross product of two vectors can also be thought of as a second order tensor times the second vector...

In this chunk, we connect the concept of tensor multiplication back to a familiar vector operation: the cross product. It shows that the cross product can be understood in the framework of tensors by interpreting one vector as a skew-symmetric tensor that converts another vector through the matrix multiplication. This connection consolidates knowledge from both vectors and tensors, enhancing understanding.

Consider teamwork where two individuals contribute their unique skills (vectors) to solve a problem (cross product). The outcome (resulting vector) takes into account the input from both, just as multiplying tensors involves using their properties together to create a new entity. The blend of different skills leads to a solution that neither could achieve independently.

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