5. Show that if |>2 +v2 , the orbit of the critical point of Fa escapes to infinity. Show that, if A Z 0. the lo...
2. A minimal surface is by definition a critical point of the area functional. Show that the graph of the function z(x, y)tany is a minimal surface. (This surface is called the helicoid.)
2. A minimal surface is by definition a critical point of the area functional. Show that the graph of the function z(x, y)tany is a minimal surface. (This surface is called the helicoid.)
A conductor wire extends from 0 to infinity on the z-axis and passes through a 5 A current. The wire of the conductor B is a circle with a radius r = 2 at the center (3,4,0) point and in the yz plane. Find the total force of wire A on wire B.
Question 2. In this exercise, you will show that Z[V-5] is not a U.F.D. (but it is an I.D., as you proved last lecture!) You will learn a common trick for reasoning about irreducibility and primality in a ring - with the help of special multiplicative functions to Z>. (i) First, calculate the units in Z[V-5] [Hint: calculate inverses first, assuming you can divide ("work- ing in Q[V-5]", and then see which ones actually lie in Z[V-5]] (ii) Next, we...
25. The point (0, 0) is the only critical point of the function f(,y)(2+y)++3 in the interior of the disk D +(+1 s 9/4). The following graph is of z .) restricted to the boundary of D, which is parameterized by -cost, y1n t, for 0s t s 2. Find the absolute minimum value of f(x, y) on D. (A) 0 (B) 2 (C) 3 25 3.25 (E) 2.9 (F) 3 (G) 3.25 (H) 4 25.. -8.25 (J) 9.25 3.75...
Describe by words and/or pictures, z, z1, z2, such that: 1) |z+5| = 3 2) -3 < Re(z) < 5 3) Arg(z1) = Arg(z2) 4) |z1| = |z2| 5) Im(z1 z*2) = 0 6) |z| = |z*| ** z* = "complex conjugate of z"
Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?
Find the values of x, y and z that correspond to the critical point of the function z = f(x,y) 3x2 + 5x + 5y + 2y?: = Enter your answer as a number (like 5, -3, 2.2) or as a calculation (like 5/3, 2^3, 5+4). T= Preview y= Preview z= Preview License Points possible: 10 Unlimited attempts.
0} Use the Division Algorithm and the fact that ν 2 is irrational to show that any element of J is a multiple of x2-2 and thus Let J Q z p V2
0} Use the Division Algorithm and the fact that ν 2 is irrational to show that any element of J is a multiple of x2-2 and thus Let J Q z p V2
8. 4 marks] (a) Suppose that a is the reflection across the line z y +2-0 and β is the reflection ac oss the line y = 2. i. Show that the compositions a(β(z)) and β(a(z)) have a common fixed point in the intersection of two lines. Find this point. ii. Find complex numbers a, b, c, and d for which a(z) = a5+b and β(z) =cz+d. iii. Classify the compositions a(β(z)) and β(a(z)). (b) Let z = 21 and...
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....