Question

Let⇀H=〈−y(2 +x), x, yz〉

(a) Show that ⇀∇·⇀H= 0.

(b) Since⇀H is defined and its component functions have continuous partials on R3, one can prove that there exists a vector field ⇀F such that ⇀∇×⇀F=⇀H. Show that F = (1/3xz−1/4y^2z)ˆı+(1/2xyz+2/3yz)ˆ−(1/3x^2+2/3y^2+1/4xy^2)ˆk

satisfies this property.

(c) Let⇀G=〈xz, xyz,−y^2〉. Show that⇀∇×⇀G is also equal to⇀H.

(d) Find a function f such that⇀G=⇀F+⇀∇f.

Answer #1

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

Let f(x, y) = −x 3 + y 2 . Show that (0, 0) is a saddle point.
Note that you cannot use the second derivative test for this
function. Hint: Find the curve of intersection of the graph of f
with the xz-plane.

(3)If H(x, y) = x^2 y^4 + x^4 y^2 + 3x^2 y^2 + 1, show that H(x,
y) ≥ 0 for all (x, y). Hint: find the minimum value of H.
(4) Let f(x, y) = (y − x^2 ) (y − 2x^2 ). Show that the origin
is a critical point for f which is a saddle point, even though on
any line through the origin, f has a local minimum at (0, 0)

Use
Gaussian Elimination to solve and show all steps:
1. (x+4y=6)
(1/2x+1/3y=1/2)
2. (x-2y+3z=7)
(-3x+y+2z=-5)
(2x+2y+z=3)

Evaluate the outward flux ∫∫S(F·n)dS of the vector
fieldF=yz(x^2+y^2)i−xz(x^2+y^2)j+z^2(x^2+y^2)k, where S is the
surface of the region bounded by the hyperboloid x^2+y^2−z^2= 1,
and the planes z=−1 and z= 2.

Let f(x, y) = x^3 − 4xy^2 , x, y ∈ R. Use the definition of
differentiability to show that f(x, y) is differentiable at (2,
1).

Let h(x,y)=k*x^2+6xy+14*y^2+4y+10.
(a) Find the minimal value of the function h for k = 2.
(b) Using envelope theorem find approximate minimal value of h
for k = 1.98.

let let T : R^3 --> R^2 be a linear transformation defined by
T ( x, y , z) = ( x-2y -z , 2x + 4y - 2z) a give an example of two
elements in K ev( T ) and show that these sum i also an element of
K er( T)

Let F(x,y,z) = yzi + xzj + (xy+2z)k
show that vector field F is conservative by finding a function f
such that
and use that to evaluate
where C is any path from (1,0,-2) to (4,6,3)

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