Applications
53–56. Ideal flow A two-dimensional
a. Verify that the curl and divergence of the given field is zero.
b. Find a potential function φ and a stream function ψ for the field.
c. Verify that φ and ψ satisfy Laplace’s equation
55.
Want to see the full answer?
Check out a sample textbook solutionChapter 14 Solutions
Calculus: Early Transcendentals (2nd Edition)
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Glencoe Math Accelerated, Student Edition
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Calculus & Its Applications (14th Edition)
- Q1: Consider a surface of revolution generated by a Bezier curve defined by the four control points P(0,0,0), (2,1,2), P₂ (3,3,0) P, (2,2,3) respectively, rotating about the X-axis. Calculate the point on the surface at u = 0.25, 0= π/2.arrow_forward2. Heat conduction in a square plate Three sides of a rectangular plate (@ = 5 m, b = 4 m) are kept at a temperature of 0 C and one side is kept at a temperature C, as shown in the figure. Determine and plot the ; temperature distribution T(x, y) in the plate. The temperature distribution, T(x, y) in the plate can be determined by solving the two-dimensional heat equation. For the given boundary conditions T(x, y) can be expressed analytically by a Fourier series (Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, 1993):arrow_forward5. The derivative is the gradient of the tangent to the function. In the case of a semi-circle, this means that the gradient vector (1, dy/d.r) should be orthogonal to the vector(arn, Y0). Verify this using the dot product and also by plotting both vectors (normalise the gradient vector).arrow_forward
- 4arrow_forwardElectromagnetic Pulse propagating at oblique angle to a dielectric interface Consider a gaussian wave pulse propagating along the z-axis from region 1 with refractive index n1 and onto a dielectric interface y = m z (for all x). To the left of this dielectric interface, the refractive index is n2. Devise an initial value computer algorithm to determine the time evolution of the reflected and transmitted electromagnetic fields for this pulse. e.g., n1 = 1 , n2 = 2 initial profile (t = 0, with z0 < 0) Ex = E0 exp[-a (z-z0)^2] By = n1 * Ex Choose parameters so that the pulse width is at least a fact of 8 less than the z- domain of integration ( -L < z < L). For the slope of the interface, one could choose m = 1.arrow_forward3. a. Given a potential function/- Ax'yz' - Br"y'z -Cx'y'z, where the coefficients 4.B.c, are the last three digits of your index number, find the conservative vector fieldF. i) Use Fin (3.a.) to calculate DİVF ii) Use Fin (3.a.) to calculate CurlF b. A conservative vector field is given as Find the potential functionarrow_forward
- Find the solution.arrow_forwardWhat is the implicit equation of the plane through 3D points (1,0,0), (0, 1, 0), and (0, 0, 1)? What is the parametric equation? What is the normal vector to this plane?arrow_forwardSuppose that a parachutist with linear drag (m=50 kg, c=12.5kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 220 m/s relative to the ground. a) Write a system of four differential equations for x,y,vx=dx/dt and vy=dy/dt. b) If theinitial horizontal position is defined as x=0, use Euler’s methods with t=0.4 s to compute the jumper’s position over the first 40 s. c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open.arrow_forward
- 2. The Lorenz equations originating from models of atmospheric physics are given as follows: dr = 10 (y - 2) dt (2a) %3D dy 28r – y -rz (2b) dt dz ay - 2.6666672 (2c) dt with initial conditions r(0) = y(0) = 2(0) = 5. (a) Evaluate the eigenvalues of the Jacobian matrix at t = 0. Is the problem stiff? Estimate the maximum time step that can be selected to keep the solution stable when the fourth-order Runge-Kutta method is used. (b) Solve the given system to t = 50 using the fourth-order Runge-Kutta method. Set the time step to 0.1. Plot the solution. All three functions (2(t), y(t), z(t)) should be present on one plot. • Set the time step to 10 3 and 10 6. Plot r(t) obtained at the three time steps (the first one is 0.1 from above) on one plot. Describe the behaviour. How does the value of the time step affect the result? Set the time step to 10-6 and use the initial conditions r(0) = y(0) = 5.0 and 2(0) = 5.00001. Plot z(t) obtained at the two different sets of initial conditions on…arrow_forward3) Simplify the following Boolean functions, using K-maps. F(w, x, y, z) = (1, 3, 4, 5, 6, 7, 9, 11, 13, 15)arrow_forward9. If p(x) = 4x – 2 and q(x) = -x + 3, as a function of x, find: (a) (p o q)(x) and determine its domain and range. (b) (q • p)(x) and determine its domain and range.arrow_forward
- C++ for Engineers and ScientistsComputer ScienceISBN:9781133187844Author:Bronson, Gary J.Publisher:Course Technology PtrOperations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole