Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
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))...
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...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P rty. I 5. [16...
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,...
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C (P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
Let F(x,y,z) = 4i – 3j + 5k and S be the surface defined by z= x2 + y2 and 22 + y2 < 4. Evaluate SJ, F. nds, where n is the upward unit normal vector.
yIP is paralle FaPat P df be the point at which f has a local extreme value. Then, dt t-t Let C be expressed parametrically as rit) (x() yit, and let P(ab) (())) This means that Viab)-r (6)Vg(a b) Because r(t) is orthogonal to C. VP) is orthogonal to the line tangent to C at P The gradient is maximized VgP) is orthogonal to C at P because gixy)0 is a level curve Therefore, the two gradients are parallel Explain...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...