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About 9000 results. 1294 free access solutions
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free/or 0.5$ |
| m348 | Define F on R3 by F(x) = (0, 0, cx3)x and let M be a compact three-dimensional manifold-with-boundary with MC {x: x3 <0}. The vector field F may be thought of as the downward pressure of a fluid of density in {x: x2 < 0}. Since a fluid exerts equal pressures in all directions, we define the buoyant force on M, due to the fluid, as −∫∂M <F,n >dA. Prove the following theorem. |
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| m349 | Define f: R -> R by
f (x) = { e-x-2 x ≠ 0,. 0 x -0,}
a. Show that f is a C00 function, and f(i) (0) = 0for all . |
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| m350 | Define g, h: {x€R2} |x| ≤ 1} ->R by
g(x,y) = (x,y, √1-x2-y2),
h(x,y) = (x,y, - √1-x2-y2), |
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| m351 | Define IP: Rn x Rn ->R by IP (x, y) = <x, y>.
(a) Find D(IP) (a,b) and (IP)’ (a,b).
(b) If f,g: R -> Rn are differentiable, and h: R -> R is defined by h(t) = <f(t), g(t)>, show that hI
(a) = .
(c) If f: R -> Rn is differentiable and |f(t) = 1 for all t, show that = 0. |
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| m352 | Define when a function f: Rn -> R is independent of the first variable and find f1 (a, b) for such f. Which functions are independent of the first variable and also of the second variable? |
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| m431 | f (x,y) = xxxxx
= (log (x))(aretan (aretan(sin(eos(xy)) – log (x + y ))))), |
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| m438 | Find a counter-example to Theorem 5-2 if condition (3) is omitted.
Following the hint, consider f: (- 2π, 2π) ->R2 defined by |
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| m439 | Find expressions for the partial derivatives of the following functions:
a. f(x,y) = f (g(x)k(y), g(x) + h(y)
b. f(x,y,z) = f(g(+ y), h(y + z))
c. f(x,y,z) = f(xy, yz, zx)
d. f(x,y) = f(x,g(x), h(x,y)) |
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| m440 | Find f1 for the following (where g: R -> R is continuous):
(a) f (x, y) = g
(b) f(x, y) = g
(c) f(x,y,z)= |
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| m441 | Find the partial derivatives of f in terms of the derivatives of g and h if<br a. f(x,y) = g(x)h(y)<br b. f(x,y) = g(x) h(f)<br c. f(x,y) =g(x)<br d. f(x,y) =g(y)<br e. f(x,y) =g(x+y) |
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| m442 | Find the partial derivatives of the following functions:
a. f(x,y,z)=xy
b. f(x,y,z)=z
c. f(x,y)=sin (xsin (y))
d. f(x,y,z)= sin (x sin (y sin(z)))
e. f(x,y,z)=xy2
f. f(x,y,z)=xy=z
g. f(x,y,z)=(x +y)2
h. f(x,y)= sin(xy)
i. f(x,y)= (sin (xy)) cos(3) |
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| m443 | Find the partial derivatives of the following functions (where g: R ->R is continuous):
(a) f(x,y ) = fx+ y g
(b) f(x,y ) =fx g
(c) f(x,y ) =f xy g
(d) f(x,y ) =f(fyg)g |
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| m465 | A function f: R2 -> R is said to be independent of the second variable if for each x € R we have f (x, y1) = f (x, y2) for all y1, y2. €R Show that f is independent of the second variable if and only if there is a function f: R->R such that f(x, y) = g(x). What is f1 (a, b) in terms of g1? |
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| m466 | A function f: Rn -> R is is homogeneous of degree m if f (tx) = tmf(x) for all x and t. If f is also differentiable, show that |
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| m467 | A function f: Rn x Rm-> Rp is bilinear if for x,x1, x2 € R n, y,y1, y2 € Rm and a € R
We have,
f(ax, y) = af (x, y) = f(x, ay)
f(x1 + x2, y) = f(x1, y) + f(x2, y)
f(x, y1 +y2) = f(x, y1) + f(x, y2)
(a) Prove that if f is bilinear, then
(b) Prove that Df (a, b) (x, y) = f (a,y) + f(x,b).
(c) (Show that the formula for Dp (a, b) in theorem 2-3 is a special case of (b |
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| m469 | Generalize the divergence theorem to the case of an -manifold with boundary in Rn. |
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| m470 | Generalize Theorem 5-6 to the case of an oriented (n - 1) -dimensional manifold in Rn.
The generalization is w Є A n-1(Mx) defined by |
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| m474 | Give an example of a bounded set C of measure 0 such that ∫ AXC does not exist. |
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| m495 | If Ί: Rn -> Rn is a norm preserving linear transformation and M is a k-dimensional manifold in Rn, show that M has the same volume as Ί(M). |
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| m498 | If A = [a1, b1] x . x [an, bn] and f: A -> R is continuous, define f: A -> R by |
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