Need help with Gradient in Polar Coordinates

In summary, the conversation is about finding the charge density in an electromagnetism problem using the formula E=kr^3 r^ and the equation gradient E=p/e0. The individuals in the conversation discussed how they got the gradient of E and the answer p=5kEor^2. They also mentioned the difference between gradient and divergence, as well as the formula for divergence in spherical coordinates.
  • #1
leonne
191
0

Homework Statement


Well the problem is a electromagnetism problem: I need to find the charge density. Given
E= kr^3 r^


Homework Equations


formula is gradient E=p/e0


The Attempt at a Solution


They got the gradient of E to be 1/r^2 (d/dr) (r^2 Er) i have no idea how they did it i know gradient is d/dx+ d/dy+d/dz so idk how they got that from E

then they got Eo(1/r^2 (d/dr)(r^2(kr^3) ... answer p=5kEor^2

Is that just a formula for any spherical polar coordinate? 1/r^2 (d/dr) (r^2 Er) Professor said something about looking in the back of the front cover for this. I just have pdf of book and doesn't have back cover. So if it is a formula can you tell me the whole thing thanks
 
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  • #2
First, ook up the difference between "gradient" and "divergence" :wink:...Then look up "divergence in spherical coordinates".
 

FAQ: Need help with Gradient in Polar Coordinates

What is the gradient in polar coordinates?

The gradient in polar coordinates is a mathematical concept used to describe the rate and direction of change of a function in a two-dimensional polar coordinate system. It is represented by a vector that points in the direction of the steepest increase of the function and its magnitude represents the rate of change.

How is the gradient calculated in polar coordinates?

The gradient in polar coordinates can be calculated using the following formula:
∇f = (∂f/∂r)er + (1/r)(∂f/∂θ)eθ
where ∂f/∂r and ∂f/∂θ represent the partial derivatives of the function with respect to the radial distance and angle, and er and eθ are the unit vectors in the radial and angular directions, respectively.

What is the significance of the gradient in polar coordinates?

The gradient in polar coordinates is important in understanding the behavior of functions in polar coordinate systems. It allows us to determine the direction and rate of change of a function, which can be useful in optimizing functions and solving problems in physics and engineering.

How is the gradient related to the directional derivative in polar coordinates?

The gradient and the directional derivative are closely related in polar coordinates. The directional derivative represents the rate of change of a function in a specific direction, while the gradient gives the direction of the steepest increase. The directional derivative in the direction of the gradient is the maximum rate of change of the function at a given point.

Can the gradient be negative in polar coordinates?

Yes, the gradient can be negative in polar coordinates. This means that the function is decreasing in the direction of the gradient. The magnitude of the gradient vector represents the rate of decrease, while the direction represents the direction of the steepest decrease. However, the gradient is always perpendicular to the level curves of the function, regardless of its sign.

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