2.3. Partial Derivative of Basis Vectors w.r.t. θ

We now show how to obtain $\frac{\partial \mathbf{e}_r}{\partial \theta}$ and $\frac{\partial \mathbf{e}_\theta}{\partial \theta}$. We again draw the z=0 plane as shown in Figure 5a. The θ coordinates of the two points A and B differ by $\Delta \theta$. Using the definition of differentiation, we can write.

In this chunk, we introduce the purpose of the section, which is to derive the partial derivatives of the basis vectors with respect to the angular coordinate θ. By setting up a coordinate system on the z=0 plane, we highlight two points A and B that have a difference in their θ coordinates denoted by the symbol Δθ. We also mention that we are using the concept of differentiation to formalize our derivation, which is crucial in understanding how the basis vectors change with angle.

Think of this situation like two points on a circular track (points A and B) that are separated by a specific angle (Δθ). This angle describes how far apart the two points are in a circular motion, much like how we quantify distance in linear terms along a straight path.

To obtain it geometrically, we parallelly transport the two radial vectors such that they start from a common point as shown in Figure 5b.

Here, we take a geometric approach to visualize how to derive the partial derivatives. We 'transport' the radial vectors of points A and B so that they originate from the same starting point. This helps in observing the angle created between the two vectors, which remains equal to Δθ. Understanding the geometric placement is critical because it allows us to explore the relationship between the basis vectors' orientation and their angular separation.

Imagine you are standing at the center of a pizza and you look at two slices (representing the vectors). As you turn your head from one slice to another, you can see how the angle between your view of the two slices corresponds to the change in θ—capturing that feeling of movement in a rotational framework.

The angle between the two radial vectors is still Δθ. The numerator in equation (7) will thus equal the third side of the triangle in Figure 5b which is also denoted by Δe there.

In this chunk, we analyze the triangle formed by the two radial vectors and their angle of separation, Δθ. The length of the third side of this triangle, which we denote as Δe, will play a key role in expressing the relationship between the basis vectors and their directional changes. Essentially, we are linking the angle to the geometric change in the basis vectors.

Consider this like three points on a triangle: two points are fixed radially while the third point represents their connection across an angle. The length of that connecting side measures how 'spread apart' two directions are, just like understanding the width of an angle in a dial on a measuring instrument.

The magnitude of the two sides representing the basis vectors has magnitude 1 as they represent unit vectors. We then draw an angle bisector to the angle Δθ.

This section brings in trigonometric concepts to further derive the change in the basis vectors. We emphasize that the basis vectors are unit vectors, meaning they have a length of 1. By constructing an angle bisector, this facilitates the understanding of how Δe behaves when Δθ gets very small, ultimately helping in calculating the partial derivatives systematically using trigonometrical limits.

You can visualize this with a garden stake set between two garden beds. As you pull the stake (representing the angle bisector) to create an even space, you're effectively finding the midpoint between the two edges of your beds—this is akin to how we equally measure changes between the two vectors around the angle.

Let us denote the direction of this vector by $\mathbf{e}$. Substituting these results in equation (7), we get the partial derivatives as $\frac{\partial \mathbf{e}_r}{\partial \theta} = \mathbf{e}_\theta$ and $\frac{\partial \mathbf{e}_\theta}{\partial \theta} = -\mathbf{e}_r$.

As we approach the limit where Δθ approaches zero, we find that the angle we derived leads us to specific values for the partial derivatives of our basis vectors. We specify that the change in the radial unit vector with respect to angle gives us the angular vector, and vice versa, reflecting the circular nature of the coordinate system. These relationships are vital for further applications in momentum balance and dynamics.

Think of this like a steering wheel in a car. The way the wheel turns (Δθ) influences the direction you go (the basis vectors). If you turn the wheel just a tiny bit, the car's trajectory changes in a specific way, showing us how sensitive these relationships are in movement.

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