Gradient in Spherical Coordinates: Computing w/ {em} & {wm}

In summary, the conversation discusses finding the gradient in spherical coordinates using the equation G-1d\phi, with \phi being a scalar function. The speaker is confused about computing in terms of {em} and {wm}, and asks if they need to convert co and contra vectors into scalars. They also mention using charts to define the gradient as a vector field on the sphere.
  • #1
autobot.d
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0
So I am working in spherical coordinates and to find the gradient I have the eqn

G-1d[tex]\phi[/tex]
where [tex]\phi[/tex] is a scalar function

Then I am supposed to compute in terms of {em} and {wm}.

I am just confused what it means to compute in terms of? Do i have to convert the
co and contra vectors into scalars? Any help is appreciated.
 
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  • #2
Just confused on what the gradient of phi in terms of the vector and covector in spherical coordinates are. Maybe that is a little easier to help with?
 
  • #3
You need to use charts to define the gradient --seen as a vector field--on the sphere.

Use the charts to pull it back from R^n.
 

Related to Gradient in Spherical Coordinates: Computing w/ {em} & {wm}

1. What are spherical coordinates?

Spherical coordinates are a system of coordinates used to represent points in three-dimensional space. They consist of a radial distance (r), an azimuth angle (θ), and a polar angle (φ).

2. How is the gradient computed in spherical coordinates?

The gradient in spherical coordinates is computed using the partial derivative with respect to each coordinate. The resulting gradient vector is given by ∇ = &fracpartial;{r}er + &frac1{r} &fracpartial;{θ}eθ + &frac1{r sin(φ)} &fracpartial;{φ}eφ, where er, eθ, and eφ are unit vectors in the r, θ, and φ directions, respectively.

3. Why is the gradient important in spherical coordinates?

The gradient is important in spherical coordinates because it represents the direction and magnitude of the steepest increase of a scalar field. This is useful in many fields of science, such as physics, engineering, and mathematics.

4. How is the gradient used in computing velocity and acceleration?

The gradient is used in computing velocity and acceleration by taking the gradient of a scalar field representing the potential or energy, and then using the resulting vector to calculate the velocity and acceleration vectors. This is commonly used in physics and engineering to model the motion of particles or fluids.

5. Can the gradient be computed in other coordinate systems?

Yes, the gradient can be computed in other coordinate systems, such as Cartesian, cylindrical, and curvilinear coordinates. Each coordinate system has its own formula for computing the gradient, but they all follow the same general concept of taking the partial derivatives with respect to each coordinate and combining them into a vector.

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