# Converting cartesian unit vectors to spherical unit vectors

• kde2520
In summary, the author is trying to show that Gauss' theorem applies to a fluid flow, but can't remember the spherical counterparts of \hat{x},\hat{y},\hat{z}. He was trying to use the dot product between \vec{u} and d\vec{A} to compute scalar product, but ended up with a virtually unintegratable expression. He thinks that he probably screwed up some algebra while doing so.
kde2520

## Homework Statement

Well, it's all in the title. I just need to show that Gauss's theorem applies to this fluid flow and have converted all my (x,y,z) components to their respective (r,theta,phi) versions, but I can't remember the spherical counterparts of $$\hat{x}$$,$$\hat{y}$$,$$\hat{z}$$.

Wouldn't it have been easier to look at the page you linked to and see that there is no direct conversion there than to assume that I didn't already do that and reply to my post?... maybe not. Thanks though.

I thought you wanted $$\hat{r}, \hat{\theta}, \hat{\phi}$$. I would call those the "spherical counterparts" of the cartesian basis vectors. What do you want?

You're right, I didn't word it very well. Here's what I was trying to compute:

$$\vec{u} \cdot d\vec{A}$$

where

$$\vec{u} = (2xy^{2}+2xz^{2})\hat{x} + (x^{2}y)\hat{y} + (x^{2}z)\hat{z}$$

and

$$d\vec{A} = r^{2}sin\theta d\theta d\phi \hat{r}$$.

I first converted the magnitudes of $$\vec{u}$$ to spherical coordinates, and what I intended to do was convert $$\hat{x}, \hat{y}, \hat{z}$$ to spherical as well. Does that make sense?

Anyway, I ended up using $$\hat{r} = sin\theta cos\phi \hat{x} + sin\theta sin\phi \hat{y} + cos\theta \hat{z}$$, leaving the unit vecors in the cartesian basis.

But then when I tried to compute the scalar product I ended up with a virtually unintegratable expression. The left side of the equation (remember I was trying to show that Gauss' theorem holds over a spherical region $$a^{2} = x^{2} + y^{2} + z^{2}$$) came out to $$\frac{8}{3} \pi a^{5}$$.

Do you think I was setting it up right? I was thinking that I probably skrewed up some algebra while computing the dot product.

If you are integrating over a sphere of radius a, then the normal vector is (x*xhat+y*yhat+z*zhat)/a. Does that suggest any simplifications? Sorry, it's kind of late here so I haven't really thought this through.

Another tip. Once you have your x,y,z expression for the integrand. You can choose the axis of the spherical coordinate system to point along any cartesian axis.

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When I converted Laplacian from cartesian to spherical I use these unit vectors. I think my weblog will be usefull for you.

Last edited by a moderator:

## 1. What are cartesian unit vectors?

Cartesian unit vectors, also known as basis vectors, are a set of three vectors (i, j, k) that are used to define a coordinate system in three-dimensional space. These vectors are perpendicular to each other and have a magnitude of 1.

## 2. What are spherical unit vectors?

Spherical unit vectors, also known as polar unit vectors, are a set of three vectors (r, θ, φ) that are used to define a coordinate system in three-dimensional space. These vectors are used in spherical coordinates, where r represents the distance from the origin, θ represents the angle between the vector and the positive z-axis, and φ represents the angle between the vector and the positive x-axis.

## 3. Why do we need to convert cartesian unit vectors to spherical unit vectors?

Converting cartesian unit vectors to spherical unit vectors allows us to represent points and vectors in three-dimensional space using a different coordinate system. This can be useful in certain situations, such as when working with polar coordinates or when solving problems involving spherical objects.

## 4. How do you convert cartesian unit vectors to spherical unit vectors?

To convert cartesian unit vectors (i, j, k) to spherical unit vectors (r, θ, φ), we can use the following equations:
r = √(x² + y² + z²)
θ = cos⁻¹(z / √(x² + y² + z²))
φ = tan⁻¹(y / x)

## 5. Can you provide an example of converting cartesian unit vectors to spherical unit vectors?

Sure, let's say we have a point P(3, 4, 5) represented in cartesian coordinates. To convert this to spherical coordinates, we would first calculate the magnitude of the point:
r = √(3² + 4² + 5²) = √50 ≈ 7.07
Next, we can calculate θ and φ using the equations mentioned in the previous answer:
θ = cos⁻¹(5 / √50) ≈ 49.04°
φ = tan⁻¹(4 / 3) ≈ 53.13°
Therefore, the point P in spherical coordinates would be (7.07, 49.04°, 53.13°).

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