Compute the flux of the vector field v(x,y)=(x,y) across the circle around the origin of radius 3.
Compute the flow of the vector field v(x,y)=(x,y) along the circle around the origin of radius 3.
I know the equations to find flux and flow, but I don't know how to do it using the vector v(x,y)=(x,y).
Do you know about divergence and curl ?
I have a sneaking suspicion that the divergence and curl (no sources or sinks and no vortices with singularities at the origin) are zero and the flux and line integral around the closed surface are zero.
To compute the flux and flow of the vector field v(x, y) = (x, y) across/along the circle around the origin of radius 3, you can use the following steps:
1. Determine the equation of the circle: The circle can be represented by the equation x^2 + y^2 = r^2, where r is the radius. In this case, r = 3, so the equation is x^2 + y^2 = 9.
2. Parameterize the circle: Choose a parameterization that traces the circle. One common parameterization for a circle is using polar coordinates:
x = r * cos(theta)
y = r * sin(theta)
where theta varies from 0 to 2*pi (a full circle).
3. Compute the derivative of the parameterization: Take the derivatives of x and y with respect to theta to get dx/dtheta and dy/dtheta, respectively.
4. Evaluate the vector field on the parameterization: Substitute the values of x and y from the parameterization into the vector field v(x, y) = (x, y). This will give you a new vector field that depends on the parameter theta.
5. Calculate the dot product: Find the dot product between the vector field and the derivative of the parameterization: v(x, y) dot (dx/dtheta, dy/dtheta).
6. Integrate the dot product: Integrate the dot product over the range of theta from 0 to 2*pi. This will give you the flux/flow of the vector field across/along the circle.
Note: Flux and flow are typically defined differently, so make sure you understand whether you need to compute the flux or flow in your specific problem. The above steps apply to both cases, but the interpretation of the result may differ.