In Problem 7 and8, find a solution to the Dirichlet boundary value problem for a disk:
for the given function
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Fundamentals of Differential Equations and Boundary Value Problems
- 1. Given f(x,y) = 2x² + y² whose graph is a paraboloid. Fill in the table with the values of the directional derivative at the points (a,b) in the directions given by the unit vectors, u, v and w. u = (1, 0) --(4+2) V = W = (0, 1) (a, b) = (1, 0) (a, b) = (1, 1)| (a, b) = (1, 2) Interpret each of the directional derivatives computed in the table at point (1,0) z=2x² + y²arrow_forwardWritten Problem 3: An Acura and a BMW are driving around an empty planar parking lot, represented by the xy-plane. The trajectories of the cars are given parametrically by A(t) =(5t +3, – 21) (the Acura), B(t)=| 21 +- 15 ,t² + 2 2 33 -* (the BMW), 4 where t represents time in minutes. a) Show that these cars crash into one another at a certain time. b) Compute the slopes of the tangent lines to the cars' trajectories at the time of the crash. Give your answers in fraction form. c) If you did part (b) correctly, you should notice something geometrically interesting about the tangent lines. What do you notice, and what does it tell you about how the cars crashed (ex: did they crash head-on)?arrow_forward5. If vector T = (x + y +1)i + j - (x + y)k then T.curl(T )isarrow_forward
- Given A = (a1, a2, az) and B = (b1, b2, b3) a parametrization of the segment line joining B to A is given by (a)X(t) = ((b, – a,)t + a4, (bz – az)t + az, (bz – az)t + az) for t E [0, 1] (b)X(t) = ((a, – b,)t + b,, (az – bz)t + b2, (az – b3)t + b3) for t E [0,1] (c)X(t) = ((b, – a,)t + b,, (b, – az)t + b2, (bz – a3)t + b3) for t € [0, 1] (a) O (b) O (c)arrow_forwardQues. 3: Check whether the operator T: R2 → R? defined by T(x, y) = (y,x) is linear. Moreover if T is linear find T' if exists.arrow_forward1. Evaluate fF.d 3 dr for F(x, y, z)=(xz)i + (x²y)j-(y²z)k where C' is given by the vector function r(t) = (²+41)i + (²-4)j + 4/²k and 0 s/s1. Carrow_forward
- (a) P, Q and R are differentiable vector functions in R³ and is a scalar u. Show that dQ d[Px (QxR)] = Px[QxdR] + Px[du du du dParrow_forwardProblem 1: Let S be the part of the paraboloid z = z² + y² that lies between the planes z = 1 and z = 5. Find the function g(x, y) and the region D where S is parametrized by the vector function r(x, y) = (x, y, g(x, y)), (x,y) in D. Include a picture of S.arrow_forwardProblem 4: Find the vector function that represents the curve of intersection for the following surfaces. x² + z? = 9 and x2 + y2 + 4z² = 25arrow_forward4. Find all the points on the paraboloid z = x² + y² - 5 where the tangent plane is parallel to the plane x + 3y - z = 0.arrow_forward4. A vector normal to the paraboloid z = (x² + y°)/2 is (a) (x, y,1). (b) (2x,2y,–1). (c) (-x,-y,1).arrow_forward2. A particle is moving along the curve having the vector equation R(t) ti+t²j+ tk. Find the vectors V(t), A(t), T(t), and N(t) and the following scalars for an arbitrary value of t: ||V(t), Ar(t), An(t), and K(t).arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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