In Problems 7 and 8 , 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
- The parametric form of the solutions of the PDES Uy + u u z = 0, u (x,0) = -x is %3D x (s, t) = st + s, y(s, t) = t, z (s,t) = s æ (s, t) = -st + s, y(s, t) = -t, 2 (s, t) = -s None x (s, t) = -st + s, y (s, t) = t, z(s, t) = -sarrow_forwardThe parametric vector form of a B-spline curve was defined in the Practice Problems as x(t) = [(1 – 1)°Po+ (3t(1 – t) – 3t + 4)P| +(3t²(1 – t) + 3t + 1)p2 + t°p;] for 0arrow_forward1. 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_forward4. Consider a square service region of unit area in which travel is right angle and directions of travel are parallel to the sides of the square. Let (X, Y₁) be the location of a mobile unit and (X₂, Y₂) the location of a demand for service. The travel distance is D =Dx + Dy where Dx = |X₁ - X₂ and Dy = |Y₁ — Y2\. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for Dx and Dy is (4(1-x)(1-y), fpx.D¸y (x, y) = {4(1 – b. Define Ryx = D/Dr. Show that the pdf of Ryx is 2 3 fryx (r) = . 2 3r² 1 r, 3 1 3r3' 0, 0≤x≤ 1,0 ≤ y ≤ 1 otherwise 0 ≤r≤1 1 ≤r <∞0arrow_forward1-8. Prove that, for any differentiable vector u Eu(t), dv = Varrow_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_forwardGiven 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_forwardProblem 4: Find the vector function that represents the curve of intersection for the following surfaces. x² + z? = 9 and x2 + y2 + 4z² = 25arrow_forward1. [V] Let P be the plane through the point (2, 0, 3) that is perpendicular to the line with parameterization r(t) = (1 + 3t)i + (2 − t)i + 4tk. Find a level set equation and two different parameterizations for P.arrow_forward1. Consider the general quadratic function f(u, v) = au² + 2buv + cv². (a) Characterize all extrema of f(u, v). Give a detailed answer. (b)Assuming A < 0, find the extrema of g(r, θ) = 2A(r – 1)² + 8A(r – 1)(θ + 2) + 8A(θ + 2)² (c) How can we employ the results in (a) for a general function with continuous first- and second-order partial derivatives?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_forward1. It is sufficient to give the correct answer. a) Someone asks you to study a function f (r) for small input vectors r. What should be your first reaction? b) Give an example of a parameterized curve r(t) = (x(t), y(t)) such that t r(0) = (0, 1), r(1) = (1, 2), [r ( 1/2 )|= v2 c) Enter a continuous function f (x, y) that is not derivable at the point (0, 1) and has a local minimum with the value 2 at this point d) Is there any function f (x, y, z) such that all its level quantities are plane x + 3y + z = C and which are not lines? Enter one in this case e) Consider the function f(x, y) = y(y-x^2 ). In which areas of the planet are f positive, negative, resp 0? Draw a figure.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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