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Circulation and flux For the following
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Calculus: Early Transcendentals (2nd Edition)
- Stokes' Theorem (1.50) Given F = x²yi – yj. Find (a) V x F (b) Ss F- da over a rectangle bounded by the lines x = 0, x = b, y = 0, and y = c. (c) fc ▼ x F. dr around the rectangle of part (b).arrow_forwardEvaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =arrow_forwardNeed help with parts (d) and (e). Thank you :)arrow_forward
- Find the gradient of the functions below. a)f (x, y, z) = sin(xyz)arrow_forwardDetermine the parametric equations of the flow lines for the velocity vector field, v, where v = yi+xjarrow_forward3. Find the directional derivative of the given function f at the given point P in the direction of the given vector a. f(x,y,z) =Vx² +y° +z', P(1,2,2), a = (3,0,4) 4. Given the equation of a surface and a point P on that surface (a) use the gradient to find a vector normal to the surface at P (b) use your answer to (a) to find an equation of the tangent plane at P. -y+z 6, P(3,2,1)arrow_forward
- Quickly pleasearrow_forward1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a helix that goes up and down the z-axis. a) Find the arclength function s(t) that gives the length of the helix from t = 0 to any other t. b) Reparametrize the helix so that it has a derivative whose magnitude is always equal to 1. c) Set R = b = 1. Compute T, Ñ, and B for the helix at the point (√2/2,√2/2, π/4).arrow_forward7. Properties of position vectors Let r = xi + yj + zk and let r = |r|. a. Show that Vr = r/r. b. Show that V(r") = nr"-2r. c. Find a function whose gradient equals r.arrow_forward
- Need help with part (e). Thank you :)arrow_forwarda) Find the gradient of f(r, y, z) = e2= sin(2x)cos(2y)arrow_forwardHeat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -k∇T, which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1. T(x, y, z) = 100 + x + 2y + z;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}arrow_forward
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