Electric Potential The electric potential V at any point (x. y) is
Sketch the equipotential curves for
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Calculus
- Calculus Assume that the level surface equation x3+y3+z3+6xyz = 1 defines z implicitly as a function of x and y. Find zx(0, −1) and zy(0, −1). Use that information to find the equation of the plane tangent to the given level surface at the point corresponding to x = 0 and y = −1−1.arrow_forwardUse Stokes's Theorem to evaluate C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (cos(y) + y cos(x))i + (sin(x) − x sin(y))j + xyzk S: portion of z = y2 over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), and (0, a)arrow_forwardAngular speed Consider the rotational velocity fieldv = ⟨ -2y, 2z, 0⟩ .a. If a paddle wheel is placed in the xy-plane with its axis normalto this plane, what is its angular speed?b. If a paddle wheel is placed in the xz-plane with its axis normalto this plane, what is its angular speed?c. If a paddle wheel is placed in the yz-plane with its axis normalto this plane, what is its angular speed?arrow_forward
- (a) Find parametric equations for the portion of the cylinder x² + y² = 5 that extends between the planes z = 0 and z = 1. (b) Find parametric equations for the portion of the cylinder x² + z² = 4 that extends between the planes y = 1 and y = 3.arrow_forwardGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forwardplease do not provide solution inimage format thank you. A thin metal plate located in the center of the xy plane has a temperature T(x, y) at the point(x,y) given by T(x, y) = 100/(1 + x^2 + y^2) . (a) What is the temperature on the plate at point (1, 2), approximately? (b) At what point is the temperature as high as possible? (c) If a particle moves away from the origin, moving along the positive x axis, Will the temperature increase or decrease? (d) At what points is the temperature 50? (e) The contour lines of T are called isotherms (because all points on a of these curves have the same temperature). Sketch some isotherms of that function.arrow_forward
- A particle is traveling along a circualr path defined by x^2 + y^2 = 4, where x and y are measured in centimeters. If the particle starts at the point (2,0) and moves in a counterclockwise direction at a speed of 6cm/sec, what will be the coodeinate fo the particle's position after 10 seconds? (show detailed diagrams)arrow_forwardFlux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.arrow_forwardLaplace equations Show that if w = ƒ(u, y) satisfies the Laplace equation ƒuu + ƒyy = 0 and if u = (x2 - y2) /2 and y = xy, then w satisfies the Laplace equation wxx + wyy = 0.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning