If f is a function of x and y, find ║∇f║^2 and ∇f and in polar coordinates r and θ.
f = f(x,y)
∇f = df/dx i + df/fy j = gradient of f
|∇f|^2 = (df/dx)^2 + (df/dy)^2
for the polar form see:
https://math.mit.edu/classes/18.013A/HTML/chapter09/section04.html
To find ║∇f║^2 (the square of the magnitude of the gradient of f) and ∇f (the gradient of f) in polar coordinates, we first need to express the gradient in terms of the radial coordinate (r) and the angular coordinate (θ).
In polar coordinates, the gradient operator ∇ can be expressed as:
∇ = (∂/∂r, 1/r * ∂/∂θ)
Now, let's find ║∇f║^2:
1. Calculate the partial derivative of f with respect to r: (∂f/∂r)
2. Calculate the partial derivative of f with respect to θ: (∂f/∂θ)
3. Determine the magnitude of the gradient (∇f) in polar coordinates using the formula:
║∇f║^2 = (∂f/∂r)^2 + (1/r * ∂f/∂θ)^2
Next, let's find ∇f:
1. Express the function f in terms of polar coordinates (r, θ).
2. Calculate the partial derivative of f with respect to r: (∂f/∂r).
3. Calculate the partial derivative of f with respect to θ: (∂f/∂θ).
The gradient of f (∇f) in polar coordinates can be expressed as:
∇f = (∂f/∂r, 1/r * ∂f/∂θ)
Remember to substitute the correct values of (∂f/∂r) and (∂f/∂θ) in the formulas based on the given function f.
(Note: The specific details of the function f were not provided in the question, so you need to use the actual function to obtain the values for ║∇f║^2 and ∇f in polar coordinates.)