Kenneth Kuttler

22.5. EXERCISES 427

2. Fick’s law for diffusion states the flux of a diffusing species, J is proportional tothe gradient of the concentration, c. Write this law getting the sign right for theconstant of proportionality and derive an equation similar to the heat equation forthe concentration, c. Typically, c is the concentration of some sort of pollutant or achemical.

3. Sometimes people consider diffusion in materials which are not homogeneous. Thismeans that J = −K∇c where K is a 3× 3 matrix. Thus in terms of components,Ji = −∑ j Ki j

∂c∂x j

. Here c is the concentration which means the amount of pollutantor whatever is diffusing in a volume is obtained by integrating c over the volume.Derive a formula for a nonhomogeneous model of diffusion based on the above.

4. Let V be such that the divergence theorem holds. Show that∫

V ∇ · (u∇v) dV =∫∂V u ∂v

∂n dA where n is the exterior normal and ∂v∂n denotes the directional derivative

of v in the direction n.

5. Let V be such that the divergence theorem holds. Show that∫V

(v∇

2u−u∇2v)

dV =∫

∂V

∂u∂n−u

∂v∂n

)dA

where n is the exterior normal and ∂u∂n is defined in Problem 4.

6. Let V be a ball and suppose ∇2u = f in V while u = g on ∂V . Show that there

is at most one solution to this boundary value problem which is C2 in V and con-tinuous on V with its boundary. Hint: You might consider w = u− v where u andv are solutions to the problem. Then use the result of Problem 4 and the identityw∇

2w = ∇ · (w∇w)−∇w ·∇w to conclude ∇w = 0. Then show this implies w mustbe a constant by considering h(t) = w(t x+ (1− t)y) and showing h is a constant.Alternatively, you might consider the maximum principle.

7. Show that∫

∂V ∇×v ·ndA = 0 where V is a region for which the divergence theoremholds and v is a C2 vector field.

8. Let F (x,y,z) = (x,y,z) be a vector field in R3 and let V be a three dimensional shapeand let n= (n1,n2,n3). Show that

∫∂V (xn1 + yn2 + zn3) dA = 3× volume of V .

9. Let F = xi+yj+zk and let V denote the tetrahedron formed by the planes, x= 0,y=0,z = 0, and 1

3 x+ 13 y+ 1

5 z = 1. Verify the divergence theorem for this example.

10. Suppose f : U →R is continuous where U is some open set and for all B⊆U whereB is a ball,

∫B f (x) dV = 0. Show that this implies f (x) = 0 for all x ∈U .

11. Let U denote the box centered at (0,0,0) with sides parallel to the coordinate planeswhich has width 4, length 2 and height 3. Find the flux integral

∫∂U F ·ndS where

F = (x+3,2y,3z). Hint: If you like, you might want to use the divergence theorem.

12. Find the flux out of the cylinder whose base is x2 + y2 ≤ 1 which has height 2 of thevector field F =

(xy,zy,z2 + x

13. Find the flux out of the ball of radius 4 centered at 0 of the vector field F =(x,zy,z+ x).

22.5.10.11.12.13.EXERCISES 427Fick’s law for diffusion states the flux of a diffusing species, J is proportional tothe gradient of the concentration, c. Write this law getting the sign right for theconstant of proportionality and derive an equation similar to the heat equation forthe concentration, c. Typically, c is the concentration of some sort of pollutant or achemical.Sometimes people consider diffusion in materials which are not homogeneous. Thismeans that J = —KVc where K is a 3 x 3 matrix. Thus in terms of components,Jj,=-Z iKi ide Here c is the concentration which means the amount of pollutantor whatever is diffusing in a volume is obtained by integrating c over the volume.Derive a formula for a nonhomogeneous model of diffusion based on the above.Let V be such that the divergence theorem holds. Show that fy, V-(uVv) dV =Sov ue dA where n is the exterior normal and ge denotes the directional derivativeof v in the direction n.. Let V be such that the divergence theorem holds. Show thatOu ov2, 2 _ _[ey u—uV v) dV [, (54 i) dAwhere n is the exterior normal and gu is defined in Problem 4.Let V be a ball and suppose Vu = f in V while u = g on OV. Show that thereis at most one solution to this boundary value problem which is C* in V and con-tinuous on V with its boundary. Hint: You might consider w = u—v where u andv are solutions to the problem. Then use the result of Problem 4 and the identitywVw=V- (wVw) — Vw- Vw to conclude Vw = 0. Then show this implies w mustbe a constant by considering A(t) = w(t a+ (1 —t) y) and showing A is a constant.Alternatively, you might consider the maximum principle.Show that /5, V x v-ndA =0 where V is a region for which the divergence theoremholds and v is a C? vector field.Let F (x,y,z) = (x,y,z) be a vector field in R? and let V be a three dimensional shapeand let n = (nj,2,n3). Show that f>) (xn; + yn2 +zn3) dA = 3x volume of V.Let F =xt+yj+2k and let V denote the tetrahedron formed by the planes, x = 0, y =0,z = 0, and 5x + ay + 5z = 1. Verify the divergence theorem for this example.Suppose f : U — R is continuous where U is some open set and for all B C U whereBis a ball, {, f (a) dV = 0. Show that this implies f(a) =0 for all a € U.Let U denote the box centered at (0,0,0) with sides parallel to the coordinate planeswhich has width 4, length 2 and height 3. Find the flux integral [>,, F -ndS whereF = (x+3,2y,3z). Hint: If you like, you might want to use the divergence theorem.Find the flux out of the cylinder whose base is x* + y* < 1 which has height 2 of thevector field F = (xy, zy, +x).Find the flux out of the ball of radius 4 centered at O of the vector field F' =(x, zy,Z+x).