1.3. Representation of displacement gradient

Now, we can plug in the displacement vector given in equation (5) in the gradient definition (4) to obtain the following: (6)

In this chunk, the focus is on how to find the displacement gradient using equations that were previously discussed. We start by taking the displacement vector from the earlier equation and substituting it into the gradient definition. This is a crucial step in understanding how the displacement of a point in a cylindrical coordinate system can change and how these changes are represented mathematically.

Imagine you're measuring how far a point on a stretchy balloon moves when the balloon is inflated. The 'displacement gradient' is like determining how the distances between various points on the surface of the balloon change when it expands—the idea is to see how a small bump on the balloon's surface changes position.

For the and terms here, the basis vectors act as a constant as they change only with θ. The derivatives of basis vectors with respect to θ were derived earlier which are (7)

Here, we discuss the concept of basis vectors in the cylindrical coordinate system. The basis vectors change only with respect to the angular coordinate θ. This means that when we're calculating gradients in relation to displacement, the behavior of these vectors remains constant in the radial and vertical directions. Understanding how derivatives of these vectors interact with the displacement is essential for finding the overall gradient.

Think of it like a compass needle that always points north regardless of your position on a roundabout. As you move around, the direction the needle points remains constant, but there may be other factors affecting your position. This shows how basis vectors can be treated as constants in certain calculations.

Upon substituting them in equation (6), we get. We have got two extra terms here due to change in basis vectors. We can use the above equation to obtain the displacement gradient matrix in cylindrical coordinate system. The coefficient of the basis tensor e ⊗ e goes into ith row and jth column of the matrix to finally yield the following: (9)

In this part, the actual substitution of derivatives into the gradient equation leads us to discover additional terms that arise because of the changes in basis vectors. These extra terms are crucial; they help construct the displacement gradient matrix in the cylindrical coordinate system, illustrating how both directionality and magnitude of displacement are relevant.

Picture a team of workers laying out a complex pattern on a football field. Each worker represents a basis vector, and their movements create a beautiful design. As each worker adjusts their position, the overall pattern shifts slightly—these changes reflect the additional terms we see in the displacement gradient matrix.

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