Deriving the Unit Vector in Spherical Coordinates from Cartesian Coordinates

Understanding the relationship between Cartesian and spherical coordinates is essential in many fields of applied mathematics, physics, and engineering. This article will guide you through the steps to derive the unit vector in spherical coordinates from its counterpart in Cartesian coordinates.

Understanding Coordinate Systems

In Cartesian coordinates, a point in three-dimensional space is represented by the coordinates x,y,z. In spherical coordinates, the same point is represented by r,θ,φ, where:

r is the radial distance from the origin to the point, θ is the azimuthal angle (angle in the xy-plane from the positive x-axis), φ is the polar angle (angle from the positive z-axis).

Step 1: Conversion from Cartesian to Spherical Coordinates

The relationships between Cartesian and spherical coordinates are given by:

x r sin φ cos θ y r sin φ sin θ z r cos φ

Step 2: Finding the Unit Vector

The unit vector in the direction of a point represented in spherical coordinates can be derived by normalizing the position vector. The position vector in spherical coordinates is given by:

(mathbf{r} r begin{bmatrix} sin varphi sin varphi cos varphi end{bmatrix} )

To find the unit vector (hat{mathbf{r}}) we normalize this vector:

(hat{mathbf{r}} frac{mathbf{r}}{r} frac{1}{r} begin{bmatrix} r sin varphi cos theta r sin varphi sin theta r cos varphi end{bmatrix} begin{bmatrix} sin varphi cos theta sin varphi sin theta cos varphi end{bmatrix} )

Step 3: The Unit Vector (hat{mathbf{r}}) in Spherical Coordinates

Therefore, the unit vector (hat{mathbf{r}}) in spherical coordinates is given by:

(hat{mathbf{r}} sin varphi cos theta boldsymbol{hat{i}} sin varphi sin theta boldsymbol{hat{j}} cos varphi boldsymbol{hat{k}})

This formula expresses how the unit vector in spherical coordinates relates back to Cartesian coordinates. The derivation provided here can help you understand the relationship more clearly and allows for efficient computation in various applications.

Projection of Unit Vectors on Various Axes

Let us consider a point P with spherical coordinates r,θ,φ. We will derive the unit vectors (hat{mathbf{r}}), (hat{mathbf{theta}}), and (hat{mathbf{phi}}) in terms of these angles.

1. Unit Vector (hat{mathbf{r}})

The projection of the unit vector (hat{mathbf{r}}) on the Z-axis is (cos varphi).

The projection of (hat{mathbf{r}}) on the XY plane is (sin varphi).

The projection of the unit vector (hat{mathbf{r}}) on the X-axis and Y-axis are (sin varphi cos theta) and (sin varphi sin theta) respectively.

Thus, (hat{mathbf{r}} sin varphi cos theta boldsymbol{hat{i}} sin varphi sin theta boldsymbol{hat{j}} cos varphi boldsymbol{hat{k}}).

2. Unit Vector (hat{mathbf{theta}})

The projection of the unit vector (hat{mathbf{theta}}) on the Z-axis is (-sin varphi).

The projection of the unit vector (hat{mathbf{theta}}) on the XY plane is (cos varphi).

The projection of the unit vector (hat{mathbf{theta}}) on the X-axis and Y-axis are (cos varphi cos theta) and (cos varphi sin theta) respectively.

Hence, (hat{mathbf{theta}} cos varphi cos theta boldsymbol{hat{i}} cos varphi sin theta boldsymbol{hat{j}} - sin varphi boldsymbol{hat{k}}).

3. Unit Vector (hat{mathbf{phi}})

The projection of the unit vector (hat{mathbf{phi}}) on the Z-axis is zero.

The projection of the unit vector (hat{mathbf{phi}}) on the X-axis and Y-axis are (-sin varphi) and (cos varphi) respectively.

Therefore, (hat{mathbf{phi}} -sin varphi boldsymbol{hat{i}} cos varphi boldsymbol{hat{j}}).

These unit vectors provide a comprehensive understanding of the geometric relationship between spherical and Cartesian coordinates.