3. Representation of vectors and second order tensors in a coordinate system

We have a vector v in space (with magnitude v) and we first choose a coordinate system (e₁, e₂, e₃) such that our vector is aligned along e₁ as shown in Figure 2. Thus, representing the vector in this coordinate system, we get:

In this chunk, we discuss how to represent a vector in a given coordinate system. A vector has a specific direction and magnitude, and its representation depends on the chosen coordinate system. Initially, the vector is aligned with the e₁ axis, which helps us easily define its components. When we write a vector this way, we can visualize its position in space better.

Imagine you're in a room and you point directly towards the north. Your direction can be thought of as a vector. Depending on whether you're using a map that has north at the top or the bottom, the representation of your direction can change, yet your actual direction remains the same.

Now, choose another coordinate system (ê₁, ê₂, ê₃) as shown in Figure 2 in red. Here, ê₁ is the same as e₁ and ê₂ makes an angle of 45° with e₁. Representation of v in this new coordinate system will be:

In this chunk, we transition to a different coordinate system where one of the axes is rotated by 45 degrees. This demonstrates how a vector's representation changes with different coordinate systems. The meaning of the vector itself remains unchanged, but its components in the new system need to be recalculated, reflecting both the rotation and the magnitude of the vector.

Think of how GPS devices work. If you were to change your orientation slightly, depending on how your GPS is set up (like changing from a portrait to landscape mode on a phone), the way your location is displayed might change even though you're still standing in the same spot.

Let us try to represent equation (8) in the (e₁, e₂, e₃) coordinate system. Now, as we know that stress tensor is a second order tensor, so its representation is going to be a matrix.

Here, we discuss the representation of a second order tensor, specifically the stress tensor, in a coordinate system. Since a second order tensor can be represented as a matrix, this section emphasizes the importance of how such a mathematical structure allows us to interpret the stress experienced by materials under various conditions in terms of its components and their orientations.

Imagine a spring. When you pull it in one direction, it reacts in a certain way. The stress tensor tells us how the spring will respond to forces applied in various directions. It’s like being able to predict how a rubber band stretches and in what directions when you pull it from different angles.

Here, a general traction component signifies the following: tᵢ represents the component of traction on the ‘i’ plane along jth direction. Thus, if we want to write down the stress matrix in (e₁, e₂, e₃) coordinate system, then the first column has to be the representation of traction on plane whose normal is along the first coordinate axis (which is e₁ here).

This chunk elaborates on the components of the stress tensor represented as a matrix format. Each element corresponds to specific forces acting on a defined area, allowing engineers to understand how stress is distributed across different axes in a material. It shows how to derive the matrix representation by considering the directions of force application relative to chosen coordinate axes.

Think of a pizza slice being pulled in different directions. The toppings move according to how you stretch or compress the slice. By understanding the stress components, one can predict where the toppings might slide off or where the cheese might tear based on the angles of pull.

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