4. Evaluate loga(-3), when loga(VT) = 0.2. 13 marks) 5. The function z is given as:...
Given that loga I = 3, loga y = 4, and loga z = 5. Using the given values, find the value of 3 loga
[3 marks] d) Suppose f(x, y,z) x3yzxy +z 3; Given: x 3 cos t; y 3 sint; z=2; i. Finds ii. Evaluate it when t -0 for f(3,1,2) iii. Evaluate it when t for f (1,1,2) dt 13 marks] 3 marks]
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1 ii. Solve the partial differential equation 18xy2 + sin(2x - y) = 0 дх2ду c) i. Solve the Lagrange equation [4 Marks] az -zp + xzq = y2 where p az and q = ду [5 Marks] x ax ii. A special form of the second order partial differential equation of the function u of the two independent variables x and t is given...
where M=7 322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0 t<-M/2 M <t< - 5) Show that the Fourier transform of function f(t) sin 7 s (10 marks) au 6) Show that u = ln(x2 + xy + y2) satisfies the partial differential equation x x ди +y 2. (7 marks) au 7) Solve the partial differential equation = e-cos(x) where at du x = 0, at =tet ax at and t = 0,...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the parametric representation r(z,0) for the cylinder x2 +y2 4 in term of z (a) and 0 (2 marks) Based on (a), find the magnitude of llr, x rell for the given cylinder (b) (6 marks) 1 1+ (e) Evaluate z dS for the given S (8 marks) Hence, use the divergence theorem to evaluate f,...
Suppose Z = XY. Suppose that X = 4 ± 0.2 and Y = 3 ± 0.1. Estimate Z in the form, a ± b. Suppose Z = XY. Suppose that X = 4 ± 0.2 and Y = 3 ± 0.1. Estimate Z in the form, a ± b.
(9 marks) QUESTION 4 a) Given vector field Éx,y,z) --yº coszi - 3xy cosz j+ xy" sinzk. Show that F is a conservative vector field. (4 marks) ii) Find the potential function, f such that F=Vf. (4 marks)
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
In problems 3-5 evaluate ∫?⃗∙??⃗? using Stokes’ theorem. In each case ? is oriented counterclockwise when viewed from above. 4. F(x, y, z) = (z)i + (x2)j + (y – sin(z))k; c is the boundary of the helicoid given by Õ(r,0) =< rcos(6), rsin(0),>; Osrs 1, osos
4. Use Stokes' Theorem to evaluate F dr. F(x,y,z)-(3z,4x, 2y); C is the circle x2 + y2 4 in the xy-plane with a counterclockwise orientation looking down the positive z-axis. az az F dr-JI, (curl F) n ds and VGy, 1) Hint: use ax' dy