Each of the following four problems is worth 25 points. Explain your answers. In partic-
ular, I should be able to follow your line of thought. The handwriting should be legible.
We will discuss these problems in class and at the Review Session (Thursday Evening).
The exam is due in in class Tuesday the 12th . Make sure that you are in class for the
last three lectures.
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Balls and spheres in any dimension. Consider the n-dimensional ball Bn(R) of radius R.
It consists of all points x = (x1,x2,...,xn) in Rn such that (x1)2+(x2)2+· · ·+(xn)2 ≤ R2.
Its boundary is the (n−1)-dimensional sphere Sn−1(R) of radius R. It consists of all points
x = (x1,2,...,xn) in Rn such that (x1)2 + (x2)2 + · ·· + (xn)2 = R2.
1. (a) Draw B1(R) and S0(R), B2(R) and S1(R), and also B3(R) and S2(R).
(b) Consider the projection P from R4x,y,u,v to R2x,y by P(x,y,u,v) = (x,y). Find the
image I(R) of the map P from B4(R) to R2x,y, i.e., the set of all values of P on B4(R),
(c) For any point (a,b) in the image I(R) find the fiber at the point (a,b) of the map P
from B4(R) to R2x,y.
(d) Use this to calculate the four dimensional volume Vol4(B4) = integraltextB4 1 dV via iterated
integration.
1
2
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2. Consider the pyramid P whose base is the square in the xy-plane with corners at
(0,0,0), (1,0,0), (0,1,0), (1,1,0), and the vertex of the pyramid is at the point (0,0,1).
For the vector field F = 〈x,y2,z3〉 calculate the integral of its normal component integraltext∂P F·dvectorS
over the boundary ∂P of the pyramid P. Here, the boundary is oriented inwards (into
the pyramid).
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3. (a) Consider a force field vectorF caused by an object at the origin of the coordinate system.
Suppose that vectorF = k φ(r) vectorr for some positive function φ and some constant k.
[[The physical meaning is that the force at a point Q is either
(1) either repelling, i.e., in the direction from origin to Q (that happens if k > 0);
(2) or attracting, i.e., in the direction from Q to origin (if k < 0);
(3) and that the size of the force at Q depends only on the distance r = |vectorr| from the
origin.]]
Show that such force is conservative.
(b) Prove that the vector field F = 〈yexy sin(z), xexy sin(z), exy cos(z)〉 is conservative.
(c) Find a potential for the vector field from (b).
(d) For the vector field from (b) calculate the integral integraltextC F· dvectorr for a curve from the point
(1,0,−π/2) to the point (0,−1,π/2), given by a very complicated formula which is too
long to write here.
3
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4. For guidance read the subsection of 8.4 on Gauss’s law (starting on the page 570) in
the book.
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(GLP) The Gauss’s law in physics says that
The flux (the total flow) of the electric field out of a closed surface S is equal to the total
electric charge inside S.
The electric field E in space is the vector field given by the electric force. Gauss’s law is
a mathematical expression of the observation that if the incoming and outgoing contribu-
tions to the flow of the electric field through a closed surface S do not cancel, then there
must be some electric charge inside S that creates the imbalance.
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(GLM) The Gauss’s law in mathematics says that for any point P we can decide whether
it is inside a given closed surface S by calculating certain integral over S. The integral
uses a vector field FP which is defined on R3 except at the point P
FP(Q) = −→PQ/|−→PQ|3.
(At the point P one would have to divide by zero!) The GLM says that
integraltextintegraltext
S FP· d
vectorS is zero if the point P does not lie inside S and it is 4π if P is enclosed by S.
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Answer the questions and justify the claims below.
• (a) Let us choose a coordinate system such that the point P is the origin. Denote
the position vector of a point by vectorr and its length by r = |vectorr|.
Show that in terms of this coordinate system the vector field FP is vectorr/r3
Explain that the above statement of GLM is the same as the one in the book.
• (b) Prove GLM when P is not inside S.
• (c) Prove GLM when S is sphere with the center P.
• (d) Prove GLM when S any closed surface that encloses P.
• (e) Explain why GLP is an example of GLM.