Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

VC definitions
Grad phi (F): its direction is along the normal vector to the level surface phi, its magnitude
gives the maximum rate of change of phi
Directional derivative: differential of phi at P in the direction of the unit vector a
Incompressible/solenoidal: div v = 0
Irrotational/conservative: curl v = 0
Dissipative/non-conservative: curl v not= 0
If the vector field is conservative, then the work integral between two points is independent
of the path chosen (provided the vector field remains finite within a simple domain)
The integral only depends on the value of phi at the two points, not the path taken between
them
...

For a closed curve, the work integral = 0
Gauss’ Divergence Theorem: if S is a closed surface bounding a volume V and F is a vector
field
• Calculate div F and integrate between limits
...
Take dot product of F with n and integrate over these products which are
non-zero, with the limits of x, y & z previously calculated
Stoke’s Theorem: let C be a closed curve and S be an open surface with C as its boundary
• Calculate the integral of the curve, C: F
...
Then calculate the surface integral of the
surface: (curlF)
...
ndA = (curlF)
...
Compare these values
...
dA = F(r)
...
(ruxrv)*dudv
o Surface integral f(r)*dA = f(r)*|dA| = f(r)*|ruxrv|*dudv (including mass, SA,
moments of inertia)



---