MATH 1104 Lecture Notes - Lecture 4: Horse Length

Find the average value of f(x, y,z) = xyz over x2 + y2 + z2 2, z > 0, x > 0. y>0. Evaluate the surface integral (x,y,1) dS, where S is the portion of the paraboloid z = 1 - x2 - y2 that lies over the unit disk. Evaluate the line integral of F(x,y,z) = (x,-y2,z), where c(r) - (sin t, cos t.2t) from t = 0 to t = pi. Express as an equivalent integral in the order dydzdx: Use the transformation u - x + y, v - x - y to find where D is the region enclosed by the lines x+y=0, x+y=l, x-y= 1, x-y=4. Find the surface integral of V x where t(x,y,z) - (z - y, z + x, - x - y) over the portion of the paraboloid z - 9- x2 -y2 above the xy-plane.

Q1. Wave equation on an infinite domain Let c> 0 and consider the wave equation on, oo): 0t2 Or2 (a) By making the change of variable ξ := x-ct and η := x + ct, show that u(z,t) is a solution of if and only if u(ξ, η) is a solution of LI Hint: Using the chain rule, show that (b) By integrating (2) show that u(z,t) = F(z-ct) + G(z + ct) for arbitrary functions of F(E) and G(η) (c) Given initial conditions u(r, 0)u( andv(), show that the solution to is co(s)ds Hint: Using the initial conditions and part (b), show that for some constants a, A v(s)ds -A

Chrome File Edit View History Bookmarks People Window Help OR 100% FZ Wed 2:33 PM a E • O • Y iowa state urx XY v MATH 207 C x Y iowa state ur D Homowork txa Man 207 FXM Inbox (,282 x C Elementary LG Elementary L x YC Modern Digi x D Mathway I GXO Co www.webassign.netweb/StudentAssignment-Responses/submit?dep=16897081 29. OH 4 points LarLinAg8 1.3.024. O76 Submissions Used My Notes Water is flowing through a network of pipes (in thousands of cubic meters per hour), as shown in the figure. (Assume F , = 400 and F. = 200, If the system has an infinite number of solutions, express x1, 2, 3, Xa, X5, x6, and x7 in terms of the parameters s and t.) X3 x4 s 6 7 (a) Solve this system for the water flow represented by Xjs i = 1, 2, . . ., 7. (x 1, X2, X3, X44 , X5, x6. X7) - ( D) (b) Find the water flow when x. = x2 = 100. (1, 2, 3, X, XS, X6, 7) = ( 0) (c) Find the water flow when Xs =X7 = 0. (X1, X2, Xa, X4, X5, X6, X7) = (d) Find the water flow when xs = 1000 and x6 = 0. (X1, 2, 3, Xa, x5, x6, X7) = Need Help? Read It II Talk to a Tutor A POg O EO DJ E® , 0 0 WAM . s IS 9 -