a07.pdf
1
AMath 231 ASSIGNMENT #7: More ∇; Gauss’ & Stokes’ Thms. Spring 2014 Due on or before Thursday, July 3rd at noon in the correct drop slot. 1. Show that the velocity ~v of a particle in cylindrical coordinates is given by: ~v = dρ dt ˆ e ρ + ρ dφ dt ˆ e φ + dz dt ˆ e z 2. a) A 3-dimensional vector field in spherical coordinates can be expressed as ~ F = F r ˆ e r + F θ ˆ e θ + F φ ˆ e φ where ˆ e r , ˆ e θ , ˆ e φ are unit vectors in the direction of increas- ing r, θ, φ, respectively at the given point. Show that the field lines satisfy the differential equations: dr dt = F r , dθ dt = F θ r , dφ dt = F φ r sin θ . b) Find the field lines of the vector field ~ F (r, θ)=ˆ e r +ˆ e θ (+0 ˆ e φ ). Explain what they look like, by sketching a cross section, for example, and describing how the rest of 3-space is filled in with the curves. 3. Evaluate Z Σ Z F · n dS in the following cases. Use Gauss’ Theorem and/or symmetry whenever convenient. (a) F =(xy, y 2 + e xz 2 , sin(xy)), Σ is the boundary surface of the region Ω bounded by the parabolic cylinder z =1 - x 2 and the planes z = 0, y = 0, and y + z = 2, with outward-pointing normal vector. (b) F =(x 2 ,y 2 ,z 2 ), Σ is the unit sphere centred at the origin. 4. Evaluate R C F · dx, where F(x, y, z )=(-y 2 , x, z 2 ) and C is the curve of intersection of the plane y + z = 2 and the cylinder x 2 + y 2 = 1, oriented counterclockwise when viewed from above. 5. Use Stokes’ Theorem to compute the integral RR Σ (∇× F) · n dS where F(x, y, z )= (xz, yz, xy) and Σ is the part of the sphere x 2 + y 2 + z 2 = 4 that lies inside the cylinder x 2 + y 2 = 1 and above the xy-plane. 6.* Let D be a bounded subset of R 3 whose boundary is a piecewise smooth closed surface ∂D, with outward unit normal n. Let f be a C 1 scalar field and G be a C 1 vector field on R 3 . Prove the generalized integration by parts formula: ZZZ D ∇f · G dV = ZZ ∂D f G · n dS - ZZZ D f ∇· G dV .
Transcript of a07.pdf
-
AMath 231 ASSIGNMENT #7: More ; Gauss & Stokes Thms. Spring 2014
Due on or before Thursday, July 3rd at noon in the correct drop slot.
1. Show that the velocity ~v of a particle in cylindrical coordinates is given by:
~v =d
dte +
d
dte +
dz
dtez
2. a) A 3-dimensional vector field in spherical coordinates can be expressed as ~F =Frer + Fe + Fe where er, e, e are unit vectors in the direction of increas-ing r, , , respectively at the given point. Show that the field lines satisfy thedifferential equations:
dr
dt= Fr,
d
dt=Fr,
d
dt=
Fr sin
.
b) Find the field lines of the vector field ~F (r, ) = er + e (+0e). Explain whatthey look like, by sketching a cross section, for example, and describing how therest of 3-space is filled in with the curves.
3. Evaluate
F n dS in the following cases. Use Gauss Theorem and/or symmetry
whenever convenient.
(a) F = (xy, y2 + exz2, sin(xy)), is the boundary surface of the region bounded
by the parabolic cylinder z = 1 x2 and the planes z = 0, y = 0, and y + z = 2,with outward-pointing normal vector.
(b) F = (x2, y2, z2), is the unit sphere centred at the origin.
4. EvaluateCF dx, where F(x, y, z) = (y2, x, z2) and C is the curve of intersection
of the plane y + z = 2 and the cylinder x2 + y2 = 1, oriented counterclockwise whenviewed from above.
5. Use Stokes Theorem to compute the integral
( F) n dS where F(x, y, z) =(xz, yz, xy) and is the part of the sphere x2 +y2 +z2 = 4 that lies inside the cylinderx2 + y2 = 1 and above the xy-plane.
6.* Let D be a bounded subset of R3 whose boundary is a piecewise smooth closed surfaceD, with outward unit normal n. Let f be a C1 scalar field and G be a C1 vector fieldon R3. Prove the generalized integration by parts formula:
D
f G dV =D
fG n dS D
f G dV .
![epaper.cxnews.cnepaper.cxnews.cn/resfile/2016-11-07/A07/A07.pdf · ! ! !"#$%&’(()*+,-./ 0123&’(()4 567/ 89&’(:;?@ A9BCDEF/ GH/ I $%JKLMNOP&’(QR(STUVW TUVXYZ[\@ ]^](https://static.fdocument.pub/doc/165x107/5e74ce2373aad03bc6015e72/-a-0123a4-567-89a-a9bcdef-gh.jpg)
