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(a) Suppose that g : R → R" is a vector-valued function and that f : R" → R is a scalar field. Define the composition h : R → R by, h(t) = f (j(t)). Show that if f and g are C1 functions, then h' (t) = Vf (7(t)) · ĝ'(t). (b) Suppose a vector field F is given by the gradient of a scalar field & according to F = -V¢. Suppose also that o(t) is a field line of F. Show that (ỡ(t)) is a decreasing function of t. (c) If a particle of mass m moves in a gradient force field F = -VV, the total energy of the particle is given by, E = m||7(t)||² + V(#(t)), where 7(t) is the particle's position at time t and ü(t) = x'(t) is its velocity. Assume that Newton's second law, F = mã(t), is satisfied, where a(t) = (t) is the particle's acceleration at time t. Prove that the energy is conserved, i.e., that dE = 0. That is why a gradient force field is called sometimes called a conservative force field.

(a) Suppose that g : R → R" is a vector-valued function and that f : R" → R is a scalar field. Define the composition h : R → R by, h(t) = f (j(t)). Show that if f and g are C1 functions, then h' (t) = Vf (7(t)) · ĝ'(t). (b) Suppose a vector field F is given by the gradient of a scalar field & according to F = -V¢. Suppose also that o(t) is a field line of F. Show that (ỡ(t)) is a decreasing function of t. (c) If a particle of mass m moves in a gradient force field F = -VV, the total energy of the particle is given by, E = m||7(t)||² + V(#(t)), where 7(t) is the particle's position at time t and ü(t) = x'(t) is its velocity. Assume that Newton's second law, F = mã(t), is satisfied, where a(t) = (t) is the particle's acceleration at time t. Prove that the energy is conserved, i.e., that dE = 0. That is why a gradient force field is called sometimes called a conservative force field.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Suppose that g : R → R" is a vector-valued function and that f : R" → R is a scalar field. Define the composition h : R → R by, h(t) = f (j(t)). Show that if f and j are Cl functions, then h' (t) = Vƒ (F(t)) · ĝ'(t). (b) Suppose a vector field F is given by the gradient of a scalar field & according to F = -VÐ. Suppose also that ở(t) is a field line of F. Show that (5(t)) is a decreasing function of t. (c) If a particle of mass m moves in a gradient force field F = -VV, the total energy of the particle is given by, E = ,m||7(t)||² + V(#(t)), where 7(t) is the particle's position at time t and ü(t) = 7' (t) is its velocity. Assume that Newton's second law, F = mā(t), is satisfied, where a(t) = i'(t) is the particle's acceleration at time t. Prove that the energy is conserved, i.e., that dE = 0. That is why a gradient force field is called sometimes called a conservative force field.
Transcribed Image Text:(a) Suppose that g : R → R" is a vector-valued function and that f : R" → R is a scalar field. Define the composition h : R → R by, h(t) = f (j(t)). Show that if f and j are Cl functions, then h' (t) = Vƒ (F(t)) · ĝ'(t). (b) Suppose a vector field F is given by the gradient of a scalar field & according to F = -VÐ. Suppose also that ở(t) is a field line of F. Show that (5(t)) is a decreasing function of t. (c) If a particle of mass m moves in a gradient force field F = -VV, the total energy of the particle is given by, E = ,m||7(t)||² + V(#(t)), where 7(t) is the particle's position at time t and ü(t) = 7' (t) is its velocity. Assume that Newton's second law, F = mā(t), is satisfied, where a(t) = i'(t) is the particle's acceleration at time t. Prove that the energy is conserved, i.e., that dE = 0. That is why a gradient force field is called sometimes called a conservative force field.
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