The 1-variable chain rule (stated in class but not yet proved): Let f(x1,...,xn) be a differentiable function defined on a subset DCR", with gradient vector Vƒ (p) at each pЄ D. Let y(t) = (v1(t), ..., Yn(t)) be a differentiable path defined on some interval a < t < b, with derivative vector 7′(t) = (v(t), ..., n(t)) for each t. Suppose that y(t) = D for each t. Prove that Exercises: d fox(t) =Vf(())-7 (0) dt о (for a 0 such that for all points q in the ball B(p, r) of radius r around p we have f(q) ≤ f(p). Prove that f(p) = 0.
The 1-variable chain rule (stated in class but not yet proved): Let f(x1,...,xn) be a differentiable function defined on a subset DCR", with gradient vector Vƒ (p) at each pЄ D. Let y(t) = (v1(t), ..., Yn(t)) be a differentiable path defined on some interval a < t < b, with derivative vector 7′(t) = (v(t), ..., n(t)) for each t. Suppose that y(t) = D for each t. Prove that Exercises: d fox(t) =Vf(())-7 (0) dt о (for a 0 such that for all points q in the ball B(p, r) of radius r around p we have f(q) ≤ f(p). Prove that f(p) = 0.
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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Transcribed Image Text:The 1-variable chain rule (stated in class but not yet proved):
Let f(x1,...,xn) be a differentiable function defined on a subset DCR",
with gradient vector Vƒ (p) at each pЄ D. Let y(t) = (v1(t), ..., Yn(t)) be a
differentiable path defined on some interval a < t < b, with derivative vector
7′(t) = (v(t), ..., n(t)) for each t. Suppose that y(t) = D for each t. Prove
that
Exercises:
d fox(t) =Vf(())-7 (0)
dt
о
(for a <t<b
1. Problems from Briggs textbook on computing f(y(t)) using the
chain rule: Section 15.4, 9-18, 29, 30
2. Find the directional derivative of the function f(x, y) = x² - 3xy along
the parabola y = x² = x+2 at the point (1,2).
The remaining problems are exercises in rigorous proofs about gradient vec-
tors of n-variable functions.
3. This problem is about the intuitive, geometric properties of gradient
vectors, which were discussed in class for the special case of a 2-variable
function.
Let f R R be a differentiable function in the plane, and fix
PЄ Rn and cЄ R. Let y(t) = (Y1(t),..., Yn(t)) be a curve defined
and differentiable on some interval -r <t<r, and suppose that that
f(y(t)) = c for all t € (−r, +r), y(0) = p, and that 7'(t) + (0,0)
(intuitively, is a curve lying on the "hypersurface" that is defined by
the equation f(x) = c).
(a) Prove that the gradient vector ▼ƒ(p) is perpendicular to the tan-
gent vector (0) (intuitively, prove that f(p) is normal to the
curve f(x,y) = c at p).
Transcribed Image Text:(b) Prove that the directional derivative of f is in the direction of the
ди
curve is zero, that is, (p) = 0 where u is a unit vector in the
direction of '(t).
(c) Prove that the directional derivative of ƒ has its maximum value
in the direction of Vƒ(p).
(Hint: apply the chain rule from class on Thursday, Day 4, Sept. 12)
: R”
4. Prove that if ƒ Rn → R is a differentiable function, and that if
f(p) = 0 for all p = Rn, then f is a constant function.
5. Let f: DR be defined on DC R", and let p be an interior point of
D at which f is differentiable. Suppose that ƒ has a local maximum
at p, meaning that there exists radius r >0 such that for all points q
in the ball B(p, r) of radius r around p we have f(q) ≤ f(p). Prove
that f(p) = 0.
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