Please solve these three questions! (1) Length of graphs a) Let a path C be given by the graph of y g(x), a 3 < b, with a piecewise C1 function g : [a, b - IR. Show that the path integral of a continuous function f: IR2- R over the path C is b) Let g : [a, b] - IR be a piecewise C1 function. The length of the graph of g on (t, g(t)). Show that [a,b]...
solve number 3 2. Show that u(x, y) = e = cos 12 + y2 is harmonic on the punctured complex plane D=C\{0}, and find a harmonic conjugate v of it. 1 2 3. Reveal, what is f(x) = u(x,y) + iv(I,y) in Problem 2?
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
1) Find the arc length for the following curves. a. y2 = 4(x + 4)3, b. x= 0<x<2 1 sys2 + 4y2 2) Find the surface area resulting from the rotation of the curve about X axis a. 9x = y2 + 18, b. y = V1 + 4x, 2<x< 6 1<x<5 3) Find the surface area resulting from the rotation of the curve about th Y axis. a, y = 1- x2 0 SX S1
solve 1-84 thank you 1-82. Rewrite each of the following expressions so that your answer has no negative or fractional exponents. b. (412 c. (x27x330 a. 1654 1-83. Sketch the graph of the function y=x+2 and determine the domain and range. 1-84. What are the domain and the range for each of the following functions? Write your answers using inequality symbols if possible. b. X c. X d . 1-85. Write one or two equations to solve the following problem...
Solve the differential equations ? Problems: Solve the following differential equations, 1) V1+x?.y?-3xV/y2 -1=0.
Solve the initial value problem Please show step by step (c) (x + y)dx - (x - y)dy = 0, y(1) = 0 (d) xy' - y2 in x + y = 0, y(1) = 2
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other. 2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
Consider two random variables with joint density fY1,Y2(y1,y2) =(2(1−y2) 0 ≤ y1 ≤ c,0 ≤ y2 ≤ c 0 otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z = Y1Y2. (10 marks) . Consider two random variables with joint density fyiy(91, y2) = 2(1 - y2) 0<n<C,0<42 <c o otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z=Y Y. (10 marks)