Gradients in three dimensions Consider the following functions f, points P, and unit
- a. Compute the gradient of f and evaluate it at P
- b. Find the unit vector in the direction of maximum increase of f at P.
- c. Find the rate of change of the function in the direction of maximum increase at P.
- d. Find the directional derivative at P in the direction of the given vector.
60.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Precalculus
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
- Simplify the following Boolean Functions using the Three Variable Maps: F(x, y, z) = xy +x'y'z'+x'y z'arrow_forwardConsider the unit square [0,1]x[0,1] suppose that the upper side of the square goes down continuously at a constant speed until it reaches the X-axis and at the same time with the same speed the line x = 0 travels continuously with a center at the origin and in a clockwise direction until reaching the same one as the previous one the curve described by the intersection of these two straight lines and called quadratrix, finds a parameterization of the curve. please give correct explanationarrow_forwardMATLAB CODE Write a function iterate that takes a numerical vector v and returns the iteration/index at which the system reaches steady state. The system must remain at the same value at the end of vector for at least 2 iterations to be considered at steady state. If the system never reaches steady state, return empty vector. Assume that the vector contains at least one value.arrow_forward
- Simplify the following Boolean functions using three-variable maps. c. F(x, y, z) = ∑ (3, 5, 6, 7) d. F(A, B, C) = ∑(0, 2, 3, 4. 6)arrow_forwardA steel Column is 10m long and Ø1.2m. It carries a load of 102MN. Given the modulus of Elasticity is 256GPa. Find the stress, Strain, anddetermine how much the column is compressed. {show all the work}arrow_forwardCreate the K-maps and then simplify for the following functions: 1). F(x,y,z) = x′y′z′ + x′yz + x′yz′arrow_forward
- Create the K-maps and then simplify for the following functions: I) F(x,y,z) = x′y′z′ + x′yz′ + xy′z′ + xyz′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_forwardShow that F(x, y, z) = xy + xz + yz has the value 1 if and only if at least two of the variables x, y, and z havethe value 1.arrow_forward
- Simplify the following Boolean functions using three-variable maps. a. F(x, y, z) = ∑ (0, 1, 5, 7) b. F(x, y, z) = ∑ (1, 2, 3, 6, 7) c. F(x, y, z) = ∑ (3, 5, 6, 7) d. F(A, B, C) = ∑(0, 2, 3, 4. 6)arrow_forwardComputer Science Write a python function, NRS(fv, Jf, xo, tol), that takes in a vector-valued, multi-variable, differentiable function (fv), Jacobian of fv (Jf), initial guess of x (xo), and tolerance (tol).arrow_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_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