II2. Suppose f (x, y, z) and g(x, y, z) are continuously differentiablefunctions with f(1,2,3) = 5 and g(1,2, 3) = 7. Assume thatthe surface f(x, y, z) = 5 and the surface g(x,Y, z) = 7 have thesame tangent plane at (1,2, 3). Evaluate Vf(1,2, 3) × Vg(1, 2, 3)(remember to give reasons).

Answered: Image /qna-images/answer/623ed57a-d6e4-4189-9144-1011fba4d427.jpg

Answered: 2, z) = 5 and the surfac e at (1,2,3).…

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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H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given point. a) x² + 3x²y-z = 0 at (1,1,4) (Answ: 9x+3y-z = 8) b) z = f(x, y, z) = r²y³z at (2,1,3) (Answ: 4x - 3y -z = 2) I. In electrostatics the force (F) of attraction between two particles of opposite charge is given by (Coulomb's law) where k is a constant and r = (x, y, z). Show that F is the gradient T (Hint: ||||||(x, y, z)||). Important problem! F(r) = k₁ of P(7) ||7-1³ -k ||1| =
please answer a, b, c in detail!
Find an equation of the tangent plane to the given surface
Let z = g(r, y) = f(3 cos(ry), y + e") provided that f(3, 5) = 7, fi(3, 5) = 2, f2(3, 5) = 5. i) Find g, (0, 4). i) Find 92 (0, 4). iii) Find the equation of the tangent plane to the surface z = f(3 cos(ry), y + ev) at the point (0, 4). Türkçe: f(3, 5) = 7, fi(3,5) = 2, f2(3, 5) = 5 olmak üzere z = g(x, y) = f(3 cos(zy), y + e") olsun. i) 91 (0, 4) değerini bulunuz. in 92 (0, 4) değerini bulunuz. im) z = f(3 cos(ry), y + e*") yüzeyine (0, 4) noktasında teğet düzlemin denklemini bulunuz. O i) 20, ii) 5, ii 20x + 5y - z = 13 O i) 20, i) 5, ii) 20x - 5y - z = -1 O i) 20, i) 15, ii) 20x + 15y + z = 53 O i) 60, i) 5, ii) 60x - 5y - z = -47 O ) -40, i) 15, ii) -40x + 15y + z = -27
Let z = g(x, y) = f(3 cos(xy), y + ey) provided that f(3, 6) = 7, f₁(3, 6) = 2, f₂(3, 6) = 5. i) Find g₁ (0,5). ii) Find g₂ (0,5). iii) Find the equation of the tangent plane to the surface = f(3 cos(xy), y + ey) at the point (0,5). Türkçe: f(3,6)= 7, f₁(3,6)=2, f₂(3, 6) = 5 olmak üzere z = g(x, y) = f(3 cos(xy), y + ev) olsun. - i) 91 (0,5) değerini bulunuz. ii) 92 (0,5) değerini bulunuz. iii) z = f(3 cos(xy), y + eu) yüzeyine (0,5) noktasında teğet düzlemin denklemini bulunuz. Oi) 25, ii) 5, iii) 25x + 5y -z = 18 Oi) 25, ii) 5, iii) 25x - 5y-z = 4 Oi) 25, ii) 15, iii) 25x + 15y + z = 68 Oi) 75, ii) 5, iii) 75x - 5y-z = -57 Oi) -50, ii) 15, iii) -50x + 15y + z = -32 Oi) 75, ii) -10, iii) 75x-10y-z = -32 Oi) -25, ii) -10, iii) -25x-10y -z = 18 Oi) -50, ii) 20, iii) -50x+ 20y-z = 18
az. 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.
Let z = g(x, y) = f(3 cos(xy), y+ e™Y) provided that f(3, 6) = 6, f1(3, 6) = 2, f2(3, 6) = 5. i) Find g1 (0, 5). ii) Find g2 (0, 5). ii) Find the equation of the tangent plane to the surface z = f(3 cos(xy), y + e™y) at the point (0, 5). Türkçe: f(3, 6) = 6, f1(3, 6) = 2, f2(3, 6) = 5 olmak üzere z = g(x, y) = f(3 cos(xy), y + e=y) olsun. i) 91 (0, 5) değerini bulunuz. ii) g2 (0, 5) değerini bulunuz. iii) z = f(3 cos(ry), y + e™y) yüzeyine (0, 5) noktasında teğet düzlemin denklemini bulunuz. о) 25, i) 5, iil) 25x + 5y - z%3 19 O i) 25, ii) 5, iii) 25x - 5y - z = 7 O i) 25, ii) 15, iii) 25x + 15y + z = 69 O i) 75, ii) 5, iii) 75x - 5y - z = -56 i) -50, ii) 15, ii) -50x + 15y + z = -31 O ) 75, ii) -10, iii) 75x -10y - z = -31 о) -25, in) -10, if) -25x -10y -z%3D19 O ) -50, ii) 20, iii) -50x + 20y - z = 19
Let z = g(x, y) = f(3 cos(ry), y + e) provided that f(3, 6) = 7, fi(3, 6) = 2, f2(3,6) = 3. i) Find g1 (0, 5). ii) Find g2 (0, 5). iii) Find the equation of the tangent plane to the surface z = f(3 cos(ry), y + e=") at the point (0, 5).
Let z = g(x, y) = f(3 cos(ry), y + e) provided that f(3, 7) = 4, fi(3,7) = 2, f2(3, 7) = 4. i) Find g, (0, 6). ii) Find 92 (0, 6). ii) Find the equation of the tangent plane to the surface z = f(3 cos(xy), y + e™v) at the point (0,6). Türkçe: f(3, 7) = 4, f1(3, 7) = 2, f2(3, 7) = 4 olmak üzere z = 9(x, y) = f(3 cos(xy), y + e#") %3D olsun. i) 91 (0, 6) değerini bulunuz. ii) 92 (0, 6) değerini bulunuz. iID z = f(3 cos(ry), y + e*") yüzeyine (0, 6) noktasında teğet düzlemin denklemini bulunuz.
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:
Let z = g(x, y) = f(3 cos(xy), y + e=) provided that f(3,5) = 4, f1(3, 5) = 2, f2(3, 5) = 4. i) Find g1 (0, 4). ii) Find g2 (0, 4). iii) Find the equation of the tangent plane to the surface z = f(3 cos(xy), y + e™ ) at the point (0, 4). Türkçe: f(3, 5) = 4, f1(3,5) = 2, f2(3, 5) = 4 olmak üzere z = g(x, y) = f(3 cos(xy), y + e™v) olsun. i) 91 (0, 4) değerini bulunuz. ii) 92 (0, 4) değerini bulunuz. ii) z = f(3 cos(xy), y + e²v) yüzeyine (0, 4) noktasında teğet düzlemin denklemini bulunuz. O i) 16, ii) 4, iii) 16x + 4y - z = 12 O i) 16, ii) 4, ii) 16x - 4y - z = 4 O i) 16, ii) 12, iii) 16x + 12y + z = 44 O i) 48, ii) 4, iii) 48x - 4y - z = -36 O i) -32, ii) 12, ii) -32x + 12y + z = -20 O i) 48, ii) -8, iii) 48x -8y - z = -20 O i) - 16, ii) -8, iii) -16x -8y - z = 12 O i) -32, ii) 16, ii) -32x + 16y - z = 12
7. Let F(x, y,z)=-2yi +2xj. Use Stokes' Theorem to find F-dr where C is a circle (a) Parallel to the yz-plane, of radius a, centered at a point on the x-axis, with either orientation. (b) Parallel to the xy-plane, of radius a, centered at a point on the z-axis, oriented counterclockwise as viewed from a point on the z-axis above the circle.

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2, z) = 5 and the surfac e at (1,2,3). Evaluate = reasons).