3. We rst need to nd a vector normal to the plane, so we take the cross product of two displacement vectors:
(32,10,25)x(12,20,45)=(1, 1, 3)x(1, 2, 1):( 5,4, 3).
Now we can apply formula (2) using any of the three points:
5(:c3)+4(y+1)3(z2) =0 4:) 5:c+4
-Section : 5.5 - 3 (use Proposition 5.1 in the book),8,11,16,18 (draw the circle, write the
boundary circle equation in cartesian coordinates and then substitute polar coordinates.
You should get r^2=2r*sin(theta) or r=sin(theta) for 0<=theta
Assignment 6 Math 131
(x y)(2x + y) dx dy, where E is the region bounded by the lines y = 2x + 4,
y = 2x + 7, y = x 2 and y = x + 1, using the change of variables formula. To find the transformation T : E E, first find T 1 by
Assignment 7 Math 131
1. Evaluate the scalar line integral C f ds, where f (x, y, z) = xy + y + z and the curve C is parameterized
as x(t) = 2ti + tj + (2 2t)k for 0 t 1.
2. Evaluate the vector line integral C F ds for the vector field F =
Assignment 4 Math 131
1. Section 4.2 - problems 7, 8 or 12, 22a or cfw_23a) and b), 30, 32 (only maximum, dont do minimum).
2. Section 4.3 - problems 3, 11, 21 or 22, 23 (use Lagrange multiplier for the disks boundary, and usual
Assignment 3 Math 131
1. Find all the second mixed partials fxx , fxy , fyx , fyy for f (x, y) = ex/y xey and for f (x, y) = ln(y 2 /x).
2. Section 2.4 problem 28, and problem 30b),c). You will need the answer to 30a), which is
fx (x, y) =
Assignment 1 Math 131
1. Find the domain and range of h(x, y, z) =
2. Let f (x) = Ax for f : R2 R3 with
9(x2 +y 2 +z 2 )
Find the component functions fi (x1 , x2 ), i = 1, 2, 3 and describe the range of f .
3. problem 40
Assignment 8 Math 131
1. Section 6.2 - problems 16, 17, 22a,b.
2. Verify both sides of Greens theorem are equal for the vector field F = y 2 i + xj and the region bounded
by the square with vertices (0, 0), (1, 0), (0, 1), (1, 1).
3. Use Green
Answer number 1.
Differential punishment is a function of the collective cost and payback associated with each homicide.
Based on the relationships of the killer to the victim, it is attributed to the societal distance among them.
If the killer and victim
Assignment 5 Math 131
R 16 R e
dy dx and
(25 x2 y 2 ) dy dx
2. Section 5.2 - problem 3 but for
x dA and problems 11, 22.
R 3 R x/2
3. Evaluate 0 x2 /4 (x2 + y 2 ) dy dx.
4. Integrate f (x, y) = x + y ove
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Math 225 Homework #3: due 9/23/2016
1. Let a, b be two positive integers at least one of which is non zero.
(a) Let k be a third positive integer. Prove from the definition that gcd(a, b) =
gcd(a, b + ka):
(i) Let d = gcd(a, b): show that d|a, d|b + ka.
Math 225 Homework #1: due 9/9/2016
Let (F, +, .) be a field as given in the lectures. As mentioned in class, certain
conclusions can be inferred from these properties.
Here are some examples with proofs, followed by some exercises for you.
(i) For all a F
Math 225 Homework #4: due 9/30/2016
We proved in class that every ideal of Z is principal , that is, it has the form
nZ for some integer n.
(a) If a, b Z define (a, b) to be cfw_ma + nb | m, n Z. Prove that (a, b) is an
(b) Now suppose that at l
Math 225 Homework #2: due 9/16/2016
1. Let R be a ring with 1. We say that an element in R is a unit if v R such
that u.v = v.u = 1.
(a) Use Q1(c) to prove that if u is a unit then so is u.
(b) Let R, S be rings with 1 and let : R S be a homomorphism of r
1. Let B be an n n matrix with real entries satisfying BB t = I = B t B. Then
in particular AB is a unitary matrix, and we can apply the spectral theorem to it:
there is a unitary matrix U such that U BU = D, and the elements (= eigenv