In: Math

(1 point) Let F=−3yi+4xjF=−3yi+4xj. Use the tangential vector form of Green's Theorem to compute the circulation integral ∫CF⋅dr∫CF⋅dr where C is the positively oriented circle x2+y2=25x2+y2=25.

**Solution:** Given a vector function F:

Now, by using the tangential form of the green's theorem, we've to calculate the circulation given by:

Here C is positively oriented curve given by:

To do this, we should know the concept of Green's theorem (tangential-form).

**Green's theorem:** Let C be a piecewise smooth,
simple closed curve in the plane and D be the region inside the
enclosed by the curve C. If F is a vector field such that

Where M and N are functions of x and y and having continuous partial derivatives in the region D, then according to the green's theorem:

**...(1)**

Here, the integral is transversed in the counter-clockwise direction

Now, in our case, the vector function and curve C is given by:

Since D is the region enclosed by the curve C, therefore, it will be:

We've to calculate:

Since

Therefore,

Now, applying the tangential form of the green's theorem, we'll get:

In the graph of the region D above, we can see that the value of x and y varies between:

Therefore, our integration becomes:

Using standard integration:

We'll get:

Using standard integration:

We'll get:

**I hope it helps you!**

Use Stokes' theorem to compute the circulation
F · dr
where F =
8xyz,
2y2z,
5yz
and C is the boundary of the portion of the plane
2x + 3y +
z = 6
in the first octant. Here C is positively oriented with
respect to the plane whose orientation is upward.

Problem 1: Use Green's Theorem to evaluate the vector line
integral
∫C [?3??−?3] ??
where ? is the circle ?2+?2=1 with counterclockwise
orientation.
Problem 2: Which of the following equations represents a plane
which is parallel to the plane
36?−18?+12?=30
and which passes through the point (3,6,1) ?
a). 6?−3?+2?=3
b). 6?+3?−2?=34
c). 36?+18?−12?=204
d). 6?−3?+2?=2
e). 36?+18?+12?=228

Use the extended divergence theorem to compute the total flux of
the vector field
F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward
from the region F that lies inside the sphere x2 + y2 + z2 = 25 and
outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom
at z = −1.

Theorem: Let K/F be a field extension and let a ∈ K be algebraic
over F. If deg(mF,a(x)) = n, then
1. F[a] = F(a).
2. [F(a) : F] = n, and
3. {1, a, a2 , ..., an−1} is a basis for F(a).

Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is a
maximal ideal in F[x] if and only if p(x) is irreducible over
F.

Verify that the Divergence Theorem is true for the vector field
F on the region E. Give the flux. F(x, y, z) = xyi + yzj + zxk, E
is the solid cylinder x2 + y2 ≤ 144, 0 ≤ z ≤ 4.

Calculate integlar c F(r)dr when F =[xlny, ye^x] using Green's
theorem. R is a rectangle whose vertices are (0,1), (3,1), (3,2),
(0,2).

Let F and G~be two vector fields in R2 . Prove that
if F~ and G~ are both conservative, then F~ +G~ is also
conservative. Note: Give a mathematical proof, not just an
example.

(3) Let V be a vector space over a field F. Suppose that a ? F,
v ? V and av = 0. Prove that a = 0 or v = 0.
(4) Prove that for any field F, F is a vector space over F.
(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1,
a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...

Verify the divergence theorem for the vector ﬁeld F = 2xzi + yzj
+z2k and V is the volume enclosed by the upper hemisphere x2 + y2 +
z2 = a2, z ≥ 0

ADVERTISEMENT

ADVERTISEMENT

Latest Questions

- P7–17 Using the free cash flow valuation model to price an IPO Assume that you have...
- - Where is the smartness incorporated in the Internet? - What is the Internet end-to-end principal?...
- Name and describe the 5 types of slavery found around the globe today.
- Kohler Corporation reports the following components of stockholders’ equity at December 31, 2018. Common stock—$15 par...
- 1) Smith & Jones has been asked by Levi Manufacturing to quote on a 4-year lease...
- Wagner Industries is comparing two different capital structures. Plan I would result in 9,500 shares of...
- A medical facility was designed for 500 lab tests per day. Since this facility has been...

ADVERTISEMENT