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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 plan...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) =...
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))...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth...
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),...
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3...
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,...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
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)...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g ...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
Use the following steps to find the general equation of the plane that intersects the surface...
Use the following steps to find the general equation of the plane that intersects the surface f (x, y) ye2x-y+5 at f(-1,3): = the y Choose any vector ū, u 0, in the xy-plane that is parallel to neither the x-axis nor product to show that i is parallel to neither axis. a) axis. Use a cross Find Df1,3) b) Choose any vector v, v 0, in the xy-plane that is orthogonal to u. Use the dot product to show...
Use the following steps to find the general equation of the plane that intersects the surface...
Use the following steps to find the general equation of the plane that intersects the surface f(x, y) ye2x-y+5 at f(-1,3) Choose any vector ū, u # 0, in the xy-plane that is parallel to neither the x-axis nor the y а) axis. Use a cross product to show that u is parallel to neither axis. Find Duf-1,3) b) Choose any vector v, v *0, in the xy-plane that is orthogonal to u. Use the dot product to show that...
(1 point) A stone is thrown from a rooftop at time t 0 seconds. Its position at time t (the components are measured in meters) is given by r()-бі-50+ (24.5-49:2) k. The origin is at the base of...
(1 point) A stone is thrown from a rooftop at time t 0 seconds. Its position at time t (the components are measured in meters) is given by r()-бі-50+ (24.5-49:2) k. The origin is at the base of the bulding, which is standing on flat ground. Distance is measured in meters. The vector i points east,j points north, and k points up. (a) How high is the rooftop? meters. (b) When does the stone hit the ground? seconds (c) Where...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
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))...
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),...
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,...
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)...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
Use the following steps to find the general equation of the plane that intersects the surface f (x, y) ye2x-y+5 at f(-1,3): = the y Choose any vector ū, u 0, in the xy-plane that is parallel to neither the x-axis nor product to show that i is parallel to neither axis. a) axis. Use a cross Find Df1,3) b) Choose any vector v, v 0, in the xy-plane that is orthogonal to u. Use the dot product to show...
Use the following steps to find the general equation of the plane that intersects the surface f(x, y) ye2x-y+5 at f(-1,3) Choose any vector ū, u # 0, in the xy-plane that is parallel to neither the x-axis nor the y а) axis. Use a cross product to show that u is parallel to neither axis. Find Duf-1,3) b) Choose any vector v, v *0, in the xy-plane that is orthogonal to u. Use the dot product to show that...
(1 point) A stone is thrown from a rooftop at time t 0 seconds. Its position at time t (the components are measured in meters) is given by r()-бі-50+ (24.5-49:2) k. The origin is at the base of the bulding, which is standing on flat ground. Distance is measured in meters. The vector i points east,j points north, and k points up. (a) How high is the rooftop? meters. (b) When does the stone hit the ground? seconds (c) Where...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
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