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For this chapter the Cauchy-Schwarz inequality can be used (without a proof). (C – S) |u7v|2...
VII (5) (a) Prove the Cauchy-Schwarz inequality for vectors in R”: v•w |v||w| for all v,...
VII (5) (a) Prove the Cauchy-Schwarz inequality for vectors in R”: v•w |v||w| for all v, w ER. Also show that equality holds if and only if v = lw for some > 0. HINT: Assume, without loss of generality, that v, w # 0. Consider the non-negative function o(t) = \v – tw|2. Show that º attains a minimum at t = 6:12. Evaluate o at this point and use the fact that ¢ is non-negative to conclude. Address...
6) One of the most powerful inequalities in mathematics is the Cauchy-Bunyakovsky-Schwarz (CBS) inequality. It states:...
6) One of the most powerful inequalities in mathematics is the Cauchy-Bunyakovsky-Schwarz (CBS) inequality. It states: if u, u E R", then lü.히< 111||1| Further, equality is achieved only when ü and v are parallel, i.e. when u = cu for some constant c. Here are several problems that explore the power and scope of CBS. a) Suppose you want to maximize f(x, y, 2) = 8x + 4y +z subject to +y2 + z2 = 1. One way is...
Properties of the dot product Please help! theoretical calculus 2. Some properties of the dot product: (a) The Cauchy-Schwartz inequality: Given vectors u and v, show that lu-vl lullv1. When is this...
Properties of the dot product
Please help!
theoretical calculus
2. Some properties of the dot product: (a) The Cauchy-Schwartz inequality: Given vectors u and v, show that lu-vl lullv1. When is this inequality an equality? (Hint: Use the relationship between u-v and the angle θ between u and v.) (b) The dot product is positive definite: Show that u u 2 0 for any vector u and that u u 0 only when u-0. (c) Find examples of vectors u,...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
VII (5) (a) Prove the Cauchy-Schwarz inequality for vectors in R”: v•w |v||w| for all v, w ER. Also show that equality holds if and only if v = lw for some > 0. HINT: Assume, without loss of generality, that v, w # 0. Consider the non-negative function o(t) = \v – tw|2. Show that º attains a minimum at t = 6:12. Evaluate o at this point and use the fact that ¢ is non-negative to conclude. Address...
6) One of the most powerful inequalities in mathematics is the Cauchy-Bunyakovsky-Schwarz (CBS) inequality. It states: if u, u E R", then lü.히< 111||1| Further, equality is achieved only when ü and v are parallel, i.e. when u = cu for some constant c. Here are several problems that explore the power and scope of CBS. a) Suppose you want to maximize f(x, y, 2) = 8x + 4y +z subject to +y2 + z2 = 1. One way is...
Properties of the dot product
Please help!
theoretical calculus
2. Some properties of the dot product: (a) The Cauchy-Schwartz inequality: Given vectors u and v, show that lu-vl lullv1. When is this inequality an equality? (Hint: Use the relationship between u-v and the angle θ between u and v.) (b) The dot product is positive definite: Show that u u 2 0 for any vector u and that u u 0 only when u-0. (c) Find examples of vectors u,...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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