We consider an arbitrary sphere of radius centred at the origin and show that is constant on that sphere by taking two arbitrary points and on the sphere and showing that .
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21 Let f and g be functions from R3 to R. Suppose fis differentiable and V...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
1. Let g : R30,0,0)-R be given by g(x, y, z) 2. 3 (a) Compute Vg(x, y, z) (b) Show that V2g V (Vg) for all (x, y, 2) (c) Verify by direct calculation that (0,0,0) for any sphere S centered at the origin. d) Why do (b) and (c) not contradict the divergence theorem? 2. Let f be C2 on R3 and satisfy Laplace's equation ▽2f-0. Such functions are called harmonic. (a) Applying Green's formulas to f and g...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that 2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
Let f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the components off and g are directionally differentiable at x, and that g is such that, for all w RM, y +az) - g(y) y, w Then the composite function F(x)-g(f(x)) is directionally differentiable at x and the following chain rule holds: F, (x,d)=g'(f(x);f,(x,d)), YdER". Let f: R -R and g : R → Rbe some...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
6. Suppose that f and g are differentiable functions on an interval (a, b), and suppose that for all (a, b) we have f'(z) = g(z) and g'(z) = -f(x). Show that f2+g2 is constant.
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
Suppose that g is differentiable at x for all x ∈ R. Let f(x) = |g(x)|. Use the Chain Rule to find f′(x).
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of f(x) ?? Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of...