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- Thread starter ebolaformula
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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.

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ecastro

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stedwards

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ebolaformula said:

Yes. [tex]v = \frac{dr}{dt} + r\frac{d\theta}{dt} + r sin \theta \frac{d\phi}{dt} [/tex]

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

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ebolaformula

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But there rises another question, why does the position vector has radial component only?

Shouldn't it be r

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ecastro

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ebolaformula

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ecastro said:

Thank you!

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stedwards

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[tex]v = \frac{dr}{dt}\hat{r} + r\frac{d\theta}{dt}\hat{\theta} + r sin \theta \frac{d\phi}{dt}\hat{\phi} [/tex]

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

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.

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.

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.

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.

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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