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5.(20 pts) Consider the unit sphere and the vector w(p) = (1,0,0) assigned to the point...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.
1 1 Consider the function...
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2...
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3)....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on an inclined plane with inclination angle ? in the gravitational field g. Follow the steps below to find the velocity V of the center of mass of the sphere as a function of time if the sphere is initially at rest. Bold font represents vectors. There exists a reaction force R at the point of contact between the sphere and the plane. The equations...
2. (20 pts.) Consider the Binary Independence Model for text document. Given a vocabulary V (w.,...
2. (20 pts.) Consider the Binary Independence Model for text document. Given a vocabulary V (w., wa) of all English words (and tokens), assume that a text document is rep- resented as a vector of binary features x-(x1, ,xd)t such that xỉ 1s I if the word wi appears in the document, and r, is 0 otherwise. We want to classify text documents into c categories. Let P(wj) be the prior probability for the class w for j-1,..., c. Assume...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these...
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these three points. (a) Find the Manhattan distance between Pi and P2 (b) Find the Manhattan distance between P1 and P3. (e) Find the Manhattan distance between P2 and P3. Q2. Consider the same points in Q1 and find the Euclidean distances between the points specified in parts (a), (b), and (e). In other words, you will be doing the above question again but now...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
PHYS 1122 Homework 1 1. The figure below shows 5 sets of tiny spheres with a...
PHYS 1122 Homework 1 1. The figure below shows 5 sets of tiny spheres with a net charge on them. The spheres are equidistant along the axis. All the values of the net charge are given, except for the middle sphere, which has the same charge in all sets. Rank the sets by the magnitude of the total electrostatic force on the middle sphere, least to greatest a) b) +e c) +e +e +e d) +e +e e) +e te...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.
1 1 Consider the function...
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on an inclined plane with inclination angle ? in the gravitational field g. Follow the steps below to find the velocity V of the center of mass of the sphere as a function of time if the sphere is initially at rest. Bold font represents vectors. There exists a reaction force R at the point of contact between the sphere and the plane. The equations...
2. (20 pts.) Consider the Binary Independence Model for text document. Given a vocabulary V (w., wa) of all English words (and tokens), assume that a text document is rep- resented as a vector of binary features x-(x1, ,xd)t such that xỉ 1s I if the word wi appears in the document, and r, is 0 otherwise. We want to classify text documents into c categories. Let P(wj) be the prior probability for the class w for j-1,..., c. Assume...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these three points. (a) Find the Manhattan distance between Pi and P2 (b) Find the Manhattan distance between P1 and P3. (e) Find the Manhattan distance between P2 and P3. Q2. Consider the same points in Q1 and find the Euclidean distances between the points specified in parts (a), (b), and (e). In other words, you will be doing the above question again but now...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
PHYS 1122 Homework 1 1. The figure below shows 5 sets of tiny spheres with a net charge on them. The spheres are equidistant along the axis. All the values of the net charge are given, except for the middle sphere, which has the same charge in all sets. Rank the sets by the magnitude of the total electrostatic force on the middle sphere, least to greatest a) b) +e c) +e +e +e d) +e +e e) +e te...
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