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a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth...
Part B Please 2. Consider a point (xo, y0, 2o) in space and a vector (a, b, c). We can use this vector to define a plane as the set of all points (x,y, z) such that the vector (x - xo, y - yo, z - zo) connecting (z, y,z) with (xo, Yo, 20) is orthogonal to (a, b, c) (the normal vector to the plane). (a) Use a dot product to write the equation of a plane through...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0 Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
Please help with this question. Thank you! 1. We say p (ro. yo, 20) is a regular point for the equation F(x, y,) 0 if the equation either defines as a differentiable function f( for (, y) in a neighborhood of (ro, Vo), or defines y as a differentiable function y-g(, a) for (r, z) in a neighborhood of (ro, 2o), or defines z as a differentiable functionh(x, y) for (x, y) in a neighborhood of (ro.o). a. Suppose p...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
Theorem 2.3.1 If f is continuous on an open rectangle (a) that contains (xo yo) then the initial value problem f(a, y), y(o)yo has at least one solution on some open subinterval of (a, b) that contains ro (b) If both fand fy are continuous on R then (2.3.1) has a unique solution on some open subinterval of (a, b) that contains ro. In Exercises 1-13 find all (xo, Vo) for which Theorem 2.3.1 implies that the initial value problem...
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3 + 2t,-t) and L2 : <t,-2 + t, 7-20 a) Verify that L1 and L2 intersect, and find their point of intersection. (b) Find the equation of the plane containing L1 and L2 (2) Consider the set of all points (a, y, z) satisfying the equation 2-y2+220. Find their intersection 0 and 2-0. Use that information to sketch a with the planes y =-3,-2,-1,0,...
Problem 1 Let gi(x, y, z)-y, 92(x, y, z)z and f(x, y, z) is a differential function We introduce F(x, y, z, A, )-f(x, y, z) - Xgi(x, y, z) - Hg2(x, y, 2). ·Show that the Lagrange system for the critical points off with constraints gi (x, y, z) = 92(x,y, z)0: F(zo, yo, 20, λο, μο)-(0, 0, 0, 0, 0) is equivalent to the one-dimensional critical point equation: df dr(ro, 0, 0) = 0, 30 = 20 =...
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
A graph of f(x, y) = 11x2 – 2xy + 2y2 + 3y is displayed. The graph can be rotated with the mouse. Reset Suppose that Po = (xo, Yo, Zo) is the unique lowest point on the surface z = f(x, y). How are Xo and yo found? o by solving the set of equations w2f/ax2 = 0 and 82f|əy2 = 0 o by solving the set of equations Əflax = 0 and əfſəy = 0