Flux integrals Assume the vector field F = ( f , g ) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral ∫ C F ⋅ n d s is independent of path; that is, ∫ C F ⋅ n d s = ψ ( B ) − ψ ( A ) .
Flux integrals Assume the vector field F = ( f , g ) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral ∫ C F ⋅ n d s is independent of path; that is, ∫ C F ⋅ n d s = ψ ( B ) − ψ ( A ) .
Flux integrals Assume the vector field F = (f, g) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral
∫
C
F
⋅
n
d
s
is independent of path; that is,
∫
C
F
⋅
n
d
s
=
ψ
(
B
)
−
ψ
(
A
)
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
The figure shows the vector field
F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2).
(a) Explain why∫c F. dr has the same value for all the three curves.
(b) What is this common value?
·SoF F.Tds for the vector field F = x²i+yj along the curve x = y2 from (1,1) to (4,-2).
Evaluate
SoF
F.Tds= (Type an integer or a simplified fraction.)
Sketch some vectors in the vector field
F(x, y) = −yi + xj.
University Calculus: Early Transcendentals (4th Edition)
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