# Time derivative of 3D Spherical Coordinate

• ebolaformula
In summary, the velocity vector for a position vector in a 3D Spherical Coordinate system is obtained by taking the time derivative of the radial component only, as the polar and azimuthal angles are not included in the position vector. The position vector for this coordinate system is simply the radial component multiplied by the unit vector in the radial direction. When calculating the velocity vector, the unit vectors must also be taken into account.
ebolaformula
When we obtain the velocity vector for position vector (r, θ, φ)
Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only?
Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?

You also need to consider the other two parameters, unless ##\theta## and ##\psi## are invariant in time. It depends on the nature of your system or problem at hand.

ebolaformula said:
When we obtain the velocity vector for position vector (r, θ, φ)
Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only?
Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?

Yes. $$v = \frac{dr}{dt} + r\frac{d\theta}{dt} + r sin \theta \frac{d\phi}{dt}$$

See http://en.wikipedia.org/wiki/Spherical_coordinate_system#Kinematics, second equation.

But there rises another question, why does the position vector has radial component only?
Shouldn't it be rrθϕ? (r,θ,ϕ are unit vectors)

The position vector for the spherical coordinate system is simply ##\boldsymbol{r} = r \boldsymbol{\hat{r}}##. You cannot use ##\theta## and ##\phi## as they are in a position vector. The scalar components of a position vector should have their units as distances. The units of ##\theta## and ##\phi## are in radians or degrees.

ecastro said:
The position vector for the spherical coordinate system is simply ##\boldsymbol{r} = r \boldsymbol{\hat{r}}##. You cannot use ##\theta## and ##\phi## as they are in a position vector. The scalar components of a position vector should have their units as distances. The units of ##\theta## and ##\phi## are in radians or degrees.

Thank you!

## 1. What is the formula for calculating the time derivative of a 3D spherical coordinate?

The formula for calculating the time derivative of a 3D spherical coordinate is given by:
dC/dt = (dr/dt)er + (rdθ/dt)eθ + (rsinθdϕ/dt)eϕ
where C represents the spherical coordinate, r is the radial distance, θ is the polar angle, ϕ is the azimuthal angle, and er, eθ, and eϕ are the unit vectors in the radial, polar, and azimuthal directions, respectively.

## 2. How is the time derivative of a 3D spherical coordinate related to velocity?

The time derivative of a 3D spherical coordinate is related to velocity through the equation:
v = dr/dt er + r(dθ/dt)eθ + r(sinθdϕ/dt)eϕ
This equation represents the velocity vector in terms of the time derivatives of the spherical coordinates and the unit vectors in the corresponding directions.

## 3. Can the time derivative of a 3D spherical coordinate be negative?

Yes, the time derivative of a 3D spherical coordinate can be negative. The time derivative represents the rate of change of the coordinate with respect to time, and this rate can be positive, negative, or zero depending on the specific situation.

## 4. Does the time derivative of a 3D spherical coordinate change with the position of the point?

Yes, the time derivative of a 3D spherical coordinate can change with the position of the point. This is because the time derivatives of the spherical coordinates are influenced by the position of the point in terms of its radial distance, polar angle, and azimuthal angle.

## 5. How is the time derivative of a 3D spherical coordinate used in physics and engineering?

The time derivative of a 3D spherical coordinate is used in physics and engineering to describe the motion and velocity of objects in three-dimensional space. It is also used in the formulation of equations for physical systems involving spherical coordinates, such as in fluid mechanics and celestial mechanics.

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