(a) Derive the analogs of Formulas (12) and (13) for surfaces of the form
(b) Let
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- 5. a) Find parametric equations for the tangent line to the curve of intersection between the paraboloid z = x²+y² and the ellipsoid 4x² +y²+z² = 9 at the point (-1,1,2). Also, find a vector function that represents the curve of intersection. b) Find parametric equations for the line tangent to the curve of intersection between S₁: x² + y² = 4 and S₂ : x² + y² - z = 0 at the point (√2, √2,4). Find a vector function for the curve of intersection.arrow_forwardAssume that the second partial derivatives of the surface X = x(0, 4) are given by Xo4 = 0, X4 - e, + 2pes X00 = 0e, – 2pe,, And has the unit normal vector N = sin0e, + cos o ez. Then the second fundamental form coefficients L, M, and N of the surface are given by L-O sin -2o cos . M-0,N- sin ở + 20 cos o L = 0 sin 0, M = 0, N = sin 0 The above answer The above a nswer L= 0 sin 0, M = 0, N - 20 cos o I. - O sin 0. M- 0. N- sin 0 + 20 cos o The a bove ans wer The above a ns werarrow_forwardA parametric representation of the curve of intersection of the two surfaces x2 + 5y2 - z = 0 and z - 4y2 = 36 is given by the vector equation : r (t) = 6cosh(t) i + 2sinh(t) j + (36 - 16 sin²(t) ) K ,0sts 2n r (t) = 6cos(t) i + 2sin(t) j + (36 - 16 sin2(t) ) k , 0 st s 2n = cos(t) i + sin(t) j' + (36 - 16 sin²(t) ) k ,0 st s 2n %3D r (t) = 6cos(t) i + 6sin(t) j + 36(1 + 4 sin2(t) ) k ,O st s 2n r (t) = cos(t) i + 3sin(t) j + 36cos2(t) k ,0 st s 2Aarrow_forward
- Consider the function f(x, y) = (eª − x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a vector which is perpendicular to the level curve of f through the point (4, 2) in the direction in which f decreases most rapidly. vector = (b) Suppose v = 77 +73+ ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (4, 2). What is a? a =arrow_forwardSketch the surface z = 3x? + y? + 1 and find its linear approximation at P-(0,-1)arrow_forwardaz. Suppose F = (2xz + 3y²) a, + (4yz²) a;. (a) Calculate S[F·dS, where S is the shaded surface in Figure 1. (c) Based on your results for parts (a) and (b), what named theorem do you think is being satisfied here, if any? (b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1. az C C (0,1,1) D (0,0,0) (A ay В ax Figure 1: Figure for Problem 1.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage