In Problems 1-10 use the divergence theorem to determine the surface integral of the vector function F on the surface S. V. S S = cosesindi+sinesinoj + cosok, where 00<2T, 0≤ ≤ T F = x²i + (x + y)j + (x+y+z)k
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- 8. Given function w = f(x, y, z) = x² – 2y? + 322 (a) Find grad f(2, -1,1) (b) Find the equation of the tangent plane at the point (2, –1, 1) to the level surface f(r, y, 2) = 5. 3 (c) Find the unit vector in the direction of the largest increse of f at the point (2, –1, 1). (d) Find the largest rate of change of f at the point (2, –1, 1).The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y, (1 — z)²). (b) Evaluate the integral JS.A demonstrating clearly how you derive the surface element ds. A.ds,(5) Consider the following space curve { 2? + y? = 1, y? + 2? = 1. Find the equations of the tangent line and normal plane at the point (1,0, 1)
- 6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).4. Suppose the function f defined on the plane satisfies fax + fyy = 0 (Laplace Equation). Show that fodr-fdy = 0 for any simple closed curve C in the plane.7. Find the length of one complete graph of the hypocycloid given by r(t)=D(cos't,sin't) for 01. 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.3. Let F = (2y² + z)i + 4.ryj + «k. (a) Show whether F is conservative. (b) Using the fundamental theorem of calculus for line integrals, prove that F. dr = -T along a curve C defined by a vector function r(t) = cos ti + sin tj + tk for 04Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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