Fields Flashcards
How should you integrate a scalar field surface? (Field is F(x,y) )
Consider the integral to be the volume under the surface. Integrate between numerical bounds for the area on the outer integral, and from one bound to an arbitrary point on the curve of the boundary (expressed in terms of the outer variable).
For example, when integrating the area under a triangular shape between points (0,0), (1,0), (1,a), integrate like so: (0)/(1) (0)/(ax) F dy dx
Express the generalised formula for integration of a function of three variables
Integral = (z1)/(z2) [ (y1(z))/(y2(z) [ (x1(y,z))_/(x2(y,z)) p(x,y,z) dx] dy] dz
How can you know if a scalar potential for a field exists?
Curl of field must be zero
What is the expression for scalar potential P of field F?
F = 🔽P
How do you find a contour in a field?
Find scalar potential equation, set equal to constant
How do you find a streamline in a field?
Set Force(x)/Force(y) = dx/dy
Therefore fy/fx = dy/dx
How do you set up dA for an integral in Cartesian coordinates?
In xy plane: Adxdy
How do you set up dA in polar coordinates on the surface of a sphere?
Build up from basics in terms of r/theta/psi
On a sphere, dA = r sin(theta) d(psi) r d(theta)
= r^2 sin(theta) d(theta)d(psi)
For full surface integral of sphere, theta integrates between Pi and 0, and psi integrates between 0 and 2Pi
How do you select a volume integral dV for a sphere in polar coordinates?
Multiply dA by thickness dr
What does the Jacobian do?
It tells you that if you have a term with dxdydz, it can be replaced with drd(theta)d(psi) * r^2sin(theta)
What does Stoke’s Theorum tell you?
Stoke’s Theorum says that for a closed curve in a field, the line integral around the curve is the same as the curl of field F times dA of the surface in the curve
What is Gauss Theorum?
Gauss’ Theorum states that the integral of the divergence through a volume is equivalent to the flux through surface
(Divergence is like generation of material)
