Skip to content

Additional exercises 6

Exercise

Consider the parametric surface

r(u,v)=(2ucosv,3usinv,u2).
  1. Write the equation for x,y,z which describes this surface.
  2. Find the fundamental vector product of r.

Exercise

Consider the parametric surface

r(u,v)=((2+cosu)sinv,(2+cosu)cosv,sinu).
  1. Write the equation for x,y,z which describes this surface.
  2. Find the fundamental vector product of r.

Exercise

Let S be the bounded portion of the paraboloid x2+y2=8z which is cut off by the plane z=4. Sketch S and choose a parametric representation of this surface. Hint: a possible parametric representation is r(u,v)=(ucosv,usinv,u2/2), (u,v)T for a suitable choice of TR2.

Use this to compute the area of S.

Exercise

Let S denote the plane surface whose boundary is the triangle with vertices at (1,0,0), (0,1,0) and (0,0,1). Consider the vector field f(x,y,z)=14(xyz).

Compute the surface integral Sfn dS, where n denotes the unit normal to S which has positive z-component.

Exercise

Consider the vector field

f(x,y,z)=(x2+yzy2+xzz2+xy).

Calculate the curl and divergence of f and find

(×f)(1,2,3) and (f)(1,2,3).

Exercise

Consider the vector field

f(x,y,z)=(z+sinyz+xcosy0).

Calculate (×f)(7,π2,3) and (f)(7,π2,3).

Exercise

Let

f(x,y,z):=(yzx)

be a vector field. Let S be the portion of the paraboloid z=1x2y2 with z0 and let n be the unit normal to S with non-negative z-component. Using Stokes' Theorem (to transform the surface integral to a line integral), evaluate S(×f)n dS.

Exercise

Let VR3 be the set of (x,y,z) such that x2+y2+z225 and z3. This solid is bounded by a closed surface S which is composed of two parts: Let S1 denote the curved top part and let S2 denote the planar part.

Consider the vector field

f(x,y,z):=(xzyz1).
  1. Evaluate the surface integral S2fn dS.
  2. Evaluate the multiple integral Vf dxdydz.
  3. Evaluate the surface integral S1fn dS.

Exercise

Consider the surface S={(x,y,z):x2+y2=z,z9}. Find a parametric form for S based on polar coordinates x=rcosθ, y=rsinθ, and find the associated fundamental vector product.

Use this to find the surface integral Sfn dS, where f is the vector field

f(x,y,z)=(y20z)

and n is the unit normal to S which has positive z-component.

Exercise

Consider the the sphere S={(x,y,z):x2+y2+z2=1/4} and the vector-field

f(x,y,z)=(2x32y32z3).

Calculate Sfn dS, where n is the outgoing unit normal on S.

Hint: Use Gauss's theorem to rewrite this as a volume integral.

Exercise

Let S be the part of the surface z=1x22y2 with z0, oriented in the positive z direction. Find

I=S×fdS,

with f(x,y,z)=(x,y2,zexy).