(c) Let M be the surface with the equation x +y° +z = 9 that is above the right triangular region withvertices (0,0), (1,0) and (1,2).(i) Write down the vector function r(x, y) and domain of the parameterization for surface M.(ii) For the vector field F =ri+yj+ (x + y) k, evaluateF. dS.M(iii) For the scalar function g(x, y, z) = xz, set up but do not evaluate an iterated double integralthat gives the surface integral of g over M.

Let be the triangular region with vertices as shown in below figure. Since the region is…

Answered: (c) Let M be the surface with the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of the surface S of equation: exz + yz + 2 = 0 An equation for the tangent line to the surface S at the point P, in the direction of the vector w = (3, −2), corresponds to:
) Suppose F(x, y) = x²i + y²jand C is the line segment segment from point P = (0,-5) to Q = (-1,-1). (a) Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond tot 0 and t = 1, respective 7(t) = (b) Using the parametrization in part (a), the line integral of F along C is F- dr i = F(F()) - F'(1) dt = " dt a. with limits of integration a = and b 1 (c) Evaluate the line integral in part (b).
3. Find vector and parametric equations of the plane that contains the two intersecting lines †‚ = [3,-1,2]+s[4,0,1],s € R and I₂ = [3,-1,2]+t[4,0,2],t€R.
6. (a) Show that the curve represented by the vector function F(t) = (5t, 3t – 4, 2t² – 6) is at the intersection between a plane and a cylinder. (b) Find a vector function that represents the intersection of the sphere x² + y² + z² = 1 and the elliptic cylinder x2 + 2z2 = 1.
Find the equation of the plane that contain point A(3,1,4) and line I given parametrically as T(L) = (2,-1,3)+L(1,3,1)
Let L be the line passing through the point P(−1, 1, 5) with direction vector d=[4, −1, 0]T, and let T be the plane defined by 5x+y-2z = 25. Find the point Q where L and T' intersect. Q=(0, 0, 0)
4. (a) (b) (c) Consider f(x, y, z) = 2²ev - 2xy. Find the equation of the level surface through the point (0, 1, -1). Find a vector normal to this level surface at the point (0, 1,-1). Is the function increasing or decreasing in the direction of the vector you found in part (b)?
Find the co-ordinate on the space curve given by the position vector r(t) 5 sin(2t)i +5 cos(2t)j + 24tk, which is at a distance of 267 units along the space curve from the co-ordinate (0,5,0).
(a) For a given vector a + 0, describe the loci (the set of points 7 determined by the equations: (x, y, z)) on R³ (i) d (2—2, у+1, 2— 3) 3D 0. (ii) ||7– a|| 21. Z-. (b) A person throws a paper ball from a height of 4m from the ground. The wind is blowing and the acceleration is given by 2/3 a(t) = ( 12v3t? – 24t + (12/5P - ) -6 for t> 0. 3 7 (0), is thrown with an initial velocity of magnitude 4m/s at an angle 0 = rad with respect to the horizontal. All measurements are in SI units. me 3 rad 4 m basket
Bracket Manufacturing uses a Kanban system for a component. Daily demand is 1,100 units. Each container has a combined waiting and processing time of 0.65 days. If the container size is 100 and the alpha value (a) is 12%, how many Kanban card sets should be authorized? Round your answer up to the whole number. Kanban card set(s) What is the maximum authorized inventory? Use the rounded value from the previous question. Round your answer to the nearest whole number. unit(s)
Let L, be the line given by the intersection of two planes 2x- y+z = 1 and x – y = -2 and let L2 be the line given by parametric equation x = 2 – t L2 : y = 3 – 5t z = t If u denotes the direction vector of L1 and uz denotes the direction vector of L2. Then (a) Find the vectors u and uz. (b) Find the dot product of the vectors u and u2. (c) Find the cross product of the vectors u and u2.
(2) Write parametrizations 7(u, v) for the following surfaces (a) a sphere with radius 5 (b) a cone (c) a cylinder (3) Write the tangent vectors r, and r, for the parameterizations in the last problem.

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