R, dr and d²r and curvilinear coordinates

In summary, the vector \vec{r} cannot be written in terms of curvilinear coordinates. The second derivative, d^2\vec{r}, is not a vector but a second order tensor. In polar coordinates, \vec{r} = r \hat{r} and d\vec{r} = dr \hat{r} + r d\theta \hat{\theta}. In spherical coordinates, \vec{r} = r\hat{e_r} and applying the derivative, we get d\vec{r}=dr\hat{e_r}+r d\hat{e_r}. To find d\hat{e_r}, we use the definition of a partial derivative and arrive at
  • #1
Jhenrique
685
4
Hellow everybody!

If ##d\vec{r}## can be written in terms of curvilinear coordinates as ##d\vec{r} = h_1 dq_1 \hat{q_1} + h_2 dq_2 \hat{q_2} + h_2 dq_2 \hat{q_2}## so, how is the vectors ##d^2\vec{r}## and ##\vec{r}## in terms of curvilinear coordinates?

Thanks!
 
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  • #2
UP!?
 
  • #3
up... pliz!
 
  • #4
You can get banned from this board for "bumping" threads.

1. The vector [itex]\vec{r}[/itex] does NOT lie in the tangent plane of a surface and so cannot be written in such a way.

2. I don't know what you mean by "[itex]d^2\vec{r}[/itex]". The second derivative? Second derivatives are NOT vectors, they are second order tensors.
 
  • #5
HallsofIvy said:
2. I don't know what you mean by "[itex]d^2\vec{r}[/itex]". The second derivative? Second derivatives are NOT vectors, they are second order tensors.

In polar coordinates ##\vec{r} = r \hat{r}##

Aplying d of derivative, we have: ##d\vec{r} = dr \hat{r} + r d\theta \hat{\theta}##

Aplying d again, we have: ##d^2\vec{r} = (d^2r - d\theta^2) \hat{r} + (2 d\theta dr + r d^2 \theta)\hat{\theta}##

But, I'd like of see this result/operation in curvilinear coordinates, just this.
 
  • #6
in spherical coordinates:

[tex]\vec{r}=r\hat{e_r}[/tex]

understand [itex]r=f(r)[/itex] and [itex]\hat{e_r}=f(\theta , \phi )[/itex]

it is well understood in multivariable calculus that, given [itex]z=f(x,y)[/itex] we have [itex]dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy[/itex] (this is intuitive as well).

now by the product rule we have [tex]d\vec{r}=dr\hat{e_r}+r d\hat{e_r}[/tex]

thus it seems we have only to compute [itex]d \hat{e_r}[/itex]. by the above theorem, [itex]d \hat{e_r}=\frac{\partial \hat{e_r}}{\partial \theta}d\theta+\frac{\partial \hat{e_r}}{\partial \phi}d\phi[/itex]. a geometric illustration works best from here, but ill outline the procedure.

by definition of a partial derivative, we have [itex]\frac{\partial \hat{e_r}}{\partial \theta}=\lim_{\Delta \theta \to 0} \frac{\hat{e_r}(\theta +\Delta \theta, \phi)-\hat{e_r}(\theta , \phi)}{\Delta \theta}[/itex]. recognize (by drawing a picture perhaps) that [itex]\hat{e_r}(\theta +\Delta \theta, \phi)-\hat{e_r}(\theta , \phi)= \sin(\phi) \hat{e_\theta}\Delta \theta[/itex]. after considering the limit we arrive at simply [itex]d \hat{e_r}=\sin(\phi) \hat{e_\theta}d\theta+\frac{\partial \hat{e_r}}{\partial \phi}d\phi[/itex]

the [itex]\phi[/itex] term is conducted in a similar fashion. if you're lost in the geometry, sketch it out (it makes much more sense that way). hope this helps!
 

1. What is the difference between R, dr, and d²r?

R, dr, and d²r are all related to curvilinear coordinates, which are a way of representing a point in space using non-rectangular coordinates. R refers to the radius or distance from the origin, while dr represents an infinitesimal change in the radius, and d²r represents the second derivative or curvature in the radius. In other words, R is a single value, dr is a vector, and d²r is a tensor.

2. How are R, dr, and d²r used in physics and mathematics?

R, dr, and d²r are commonly used in physics and mathematics to describe the motion and forces acting on objects in curved spaces. They are particularly useful in fields such as general relativity, where space and time are curved, and in differential geometry, which studies the properties of curved surfaces.

3. What are some examples of curvilinear coordinates?

Some common examples of curvilinear coordinates include polar coordinates, which use a radius and angle to locate a point in a plane, and spherical coordinates, which use a radius, azimuth angle, and elevation angle to locate a point in 3-dimensional space. Other examples include cylindrical coordinates, ellipsoidal coordinates, and parabolic coordinates.

4. How do R, dr, and d²r relate to Cartesian coordinates?

In Cartesian coordinates, a point is located using its distances along three perpendicular axes (x, y, and z). In curvilinear coordinates, R represents the distance from the origin, while dr and d²r represent changes in this distance. This can be thought of as equivalent to the x, y, and z components of a vector in Cartesian coordinates.

5. What is the significance of using curvilinear coordinates?

Using curvilinear coordinates can make certain calculations and problems much simpler in cases where the underlying space is curved. For example, in physics, it allows for the description of motion and forces in non-Euclidean spaces, and in mathematics, it can simplify calculations for solving differential equations on curved surfaces.

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