# Derivative in cylindrical coordinates

Physics Asked on October 9, 2020

Why do we multiply a $$frac{1}{r}$$ factor for the gradient unit vector in $$vec{theta}$$ direction? and how is the angle a vector here?

the components of the vector $$vec{R}$$ given with polar coordinates are:

$$vec{R}= begin{bmatrix} x \ y \ end{bmatrix}= r,begin{bmatrix} cos(varphi) \ sin(varphi) \ end{bmatrix}=r,cos(varphi)hat{e}_x+r,sin(varphi)hat{e}_y$$

with :

$$hat{vec{e}_r}=frac{1}{||frac{partial vec{R}}{partial r}||},frac{partial vec{R}}{partial r}=begin{bmatrix} cos(varphi) \ sin(varphi) \ end{bmatrix}$$

and

$$hat{vec{e}_phi}=frac{1}{||frac{partial vec{R}}{partial varphi}||},frac{partial vec{R}}{partial varphi}=begin{bmatrix} -sin(varphi) \ cos(varphi) \ end{bmatrix}$$

you can write the vector R in coordinates $$quad hat{vec{e}_r},,hat{vec{e}_phi}quad$$ system

$$vec{R}=a_r,hat{vec{e}_r}+a_phi,hat{vec{e}_phi}$$

where $$a_r=r$$ and $$a_varphi=0$$

The transformation matrix between those two coordinate system is:

$$S=left[hat{vec{e}_r},,hat{vec{e}_phi}right]$$

Correct answer by Eli on October 9, 2020

The position vector (or the radius vector) is a vector R that represents the position of points in the Euclidean space with respect to an arbitrarily selected point O, known as the origin. We can parameterize this position vector as a function of coordinates and define basis vectors using it. In polar coordinates, it is a function of $$r$$ and $$theta$$, where $$r$$ is the radial distance from a chosen origin and $$theta$$, is the angle which the radial distance from the arbitrarily chosen origin makes with a line.

We can derive the basis vector of this polar coordinate system by doing derivatives on this parameterized position vector. The partial derivative of this position vector with respect to $$theta$$ gives the local basis in $$theta$$ direction. The word local is used because unlike the cartesian coordinate system, the polar coordinate system has a natural basis that changes from point to point.

Working out the angular polar basis vector:

Consider taking the partial of $$R$$ w.r.t $$theta$$ when the radial distance is some $$r$$ away, you get:

$$frac{ partial R}{partial theta} =vec{ e_{theta}}$$

But if you did this at some other $$r'$$ distance away with $$r'>r$$, your basis vector scales up by some factor. Most precisely, the basis evaluated on the unit circle scales up linearly scales a factor of $$r$$ as we move outwards from the origin and hence the unit polar basis vector at some radial distance is given as:

$$overline{vec{e_{theta}}} = frac{ e_{theta}}{r}$$

The overline stands for the vector being normalized

References:

Pavel Grinfield's Tensor Analysis book

Answered by Buraian on October 9, 2020

1. we do it since it comes out this way, if yo transform from cartesian ti polar coordinates see https://en.wikipedia.org/wiki/Polar_coordinate_system
2. in the grad you alway have 1/length so you need the 1/r to get the right dimension of grad.
3. how will you describe motion in polar coordinates if the angle has no direction?

Answered by trula on October 9, 2020

## Related Questions

### Energy in Dielectric Material

1  Asked on December 29, 2020 by user100411

### How to measure the doppler effect perpendicular to the movement in laser doppler anemometry

0  Asked on December 29, 2020

### Calculated heat capacity different (lower) from experimental value?

1  Asked on December 29, 2020 by mecury-197

### How deep would a moonbase have to be dug for radiation protection?

2  Asked on December 29, 2020 by ambrose-swasey

### How much the Earth atmosphere oscillates due to Moon tidal force and does this produces winds?

1  Asked on December 29, 2020

### Finding a “Principle of Least Action” equivalent statement for Hamiltonian Mechanics

2  Asked on December 28, 2020 by firest

### Violation of Stefan’s law when shining a light on a black body

2  Asked on December 28, 2020 by vaishakh-sreekanth-menon

### Diagonalizing a constant metric tensor $g_{munu}$ at a point

2  Asked on December 28, 2020

### I want help in recommendation of resources for my upcoming semester course on Magnetic materials fundamentals and applications

0  Asked on December 28, 2020

### With a machete, why is a diagonal cut more effective than a right angle one?

2  Asked on December 28, 2020 by user56903

### Why do 2 solenoids attract each other?

2  Asked on December 28, 2020

### Joule free expansion and differentials in irreversible processes

1  Asked on December 28, 2020 by alexk745

### Two observers initially at rest in the expanding Universe

2  Asked on December 28, 2020 by kostya

### Computing divergence of gradient in a field with measured velocity vectors

0  Asked on December 28, 2020 by albert-lidel

### Interference and Fermat’s principle

0  Asked on December 28, 2020 by snowraider

### Induced electric field ambiguities

1  Asked on December 28, 2020 by physicsa

### Intuitive argument for symmetry of Lorentz boosts

4  Asked on December 28, 2020

### What is the effect of radius in a planet’s steady state temperature?

1  Asked on December 28, 2020 by vintagelime

### Demonstration that electric current at equilibrium is zero in crystals

1  Asked on December 28, 2020 by gippo

### Does Hubble’s constant apply to galaxies that are blue-shifted/ moving towards us? Another question

1  Asked on December 28, 2020