Question 1

The work done by F(x, y) =6 x i + 6xy j along the curve x = y² from (0, 0) to (1, 1) is A) 4.5 B) 9 C) 6 D) 12 E) 0

Accepted Answer

The work done by a vector field along a curve is the line integral of the field along the path. For the given vector field F(x, y) = 6x i + 6xy j and the curve x = y^2, the line integral from (0, 0) to (1, 1) is ∫(0 to 1) [6(y^2) + 6(y^2)y] dy = ∫(0 to 1) [6y^2 + 6y^3] dy = [2y^3 + (3/2)y^4] from 0 to 1 = 2(1)^3 + (3/2)(1)^4 = 2 + 1.5 = 3.5.

Question 2

The work done by  along the curve  from (0, 0) to (1, 1) is
A) 
B) 
C) 
D) 
E)

Accepted Answer

The work done by a vector field F along a curve C is given by the line integral of F along C. In this case, the vector field F = (x^2 + y, x + y^2) and the curve C is the line segment from (0,0) to (1,1). We can parameterize this curve by r(t) = (t,t) for 0 ≤ t ≤ 1. Now, the line integral is given by:

∫(0 to 1) F(r(t)) ⋅ r'(t) dt

= ∫(0 to 1) (t^2 + t, t + t^2) ⋅ (1, 1) dt

= ∫(0 to 1) (2t^2 + 2t) dt

= 2/3

Therefore, the work done by F along C is 2/3, which is equal to choice (B).

Question 3

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin²t j; .

Accepted Answer

The answer of Find  where F(x, y) = 3(2x...

Question 4

Determine whether  is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)
A) 
B) 
C) 
D) 
E) Not conservative

Accepted Answer

The answer of Determine whether  is conservative. If it...

Question 5

The work done by  along the curve  from (0, 0) to (1, 1) is
A) 
B) 
C) 
D) 
E)

Accepted Answer

The curve from (0,0) to (1,1) can be parameterized by x=t and y=t, where t goes from 0 to 1. Then, the work done is given by the line integral:

∫(y^2-x) dx + (2xy-y)dy

Evaluating this integral along the parameterization yields:

∫t^2-t dt + (2t^2-t)dt = [t^3/3 - t^2/2 + t^2 - t^2/2] from 0 to 1 = 1/3

Thus, the work done is 11ead0bc_8d92_d300_99a0_9b2a867660b9_TB6988_11.

Question 6

Is F(x, y) = 7x i + 8y j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 7x i +...

Question 7

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y² - 3x sin y)j is conservative. If it is, find a potential function for it.

Accepted Answer

The answer of Determine whether F(x, y) = 4(3 cos...

Question 8

Is F(x, y) = 5x i + 5y j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 5x i +...

Question 9

Determine whether  is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)
A) 
B) 
C) 
D) 
E) Not conservative

Accepted Answer

The answer of Determine whether  is conservative. If it...

Question 10

Is F(x, y) = 2cos x i + 2cos y j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 2cos x i...

Question 11

Find where F(x, y) = 2(2xy² + 1)i + 4x²y j and C is the curve r(t) = e ^t sin t i + e ^t cos t j; .

Accepted Answer

The answer of Find where F(x, y) = 2(2xy²...

Question 12

Let  . Is the field conservative?

Accepted Answer

The answer of Let  . Is the field conservative?...

Question 13

Determine whether  is conservative. If it is, find a potential function for it. ( K is an arbitrary constant.)
A) 
B) 
C) 
D) 
E) Not conservative

Accepted Answer

The answer of Determine whether  is conservative. If it...

Question 14

Find the work done by the conservative force  as it acts on a particle moving from P(0, 0) to  .

Accepted Answer

The answer of Find the work done by the conservative...

Question 15

Determine whether  is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)
A) 
B) 
C) 
D) 
E) Not conservative

Accepted Answer

The answer of Determine whether  is conservative. If it...

Question 16

Determine whether F(x, y) = 5(2x + y³)i + 5(3xy² - e^-²^y )j is conservative. If it is, find a potential function for it.

Accepted Answer

The answer of Determine whether F(x, y) = 5(2x +...

Question 17

Is F(x, y) = 2y i + 2x j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 2y i +...

Question 18

Is F(x, y) = 12cos y i + 12cos x j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 12cos y i...

Question 19

For F(x, y) = 3x² i - 3y j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(3, 2) is A) 33 Nm B) 21 Nm C) -21 Nm D) -33 Nm E) 0 Nm

Accepted Answer

Using the line integral formula, we have:
W = ∫PQ F·ds
where F is the force field, ds is the differential of arc length, and the integral is taken over the curve from P to Q.

Parametrize the curve from P to Q as r(t) = 3t i + 2t j, where 0 ≤ t ≤ 1. Then, dr/dt = 3i + 2j, and ||dr/dt|| = √(3^2 + 2^2) = √13.

Substituting F and dr/dt into the line integral formula, we get:
W = ∫0^1 F(r(t)) · dr/dt dt
= ∫0^1 (9t^2 - 12t^2 + 0) i - (6t)j · (3i+2j)/√13 dt
= ∫0^1 (-3√13t^2 + 2√13t) dt
= √13 [(-t^3/3) + (t^2/2)] from 0 to 1
= √13 [(-1/3) + (1/2)]
= √13/6.

Therefore, the answer is B) 21Nm, which is approximately 3.063 Nm (rounded to three decimal places).

Question 20

Determine whether F(x, y) = 6(y² - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

Accepted Answer

The answer of Determine whether F(x, y) = 6(y² -...

Quiz 15: Topics in Vector Calculus

The work done by F(x, y) =6 x i + 6xy j along the curve x = y2 from (0, 0) to (1, 1) is

The work done by along the curve from (0, 0) to (1, 1) is

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin2t j; .

Determine whether is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)

The work done by along the curve from (0, 0) to (1, 1) is

Is F(x, y) = 7x i + 8y j is a conservative vector field?

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y2 - 3x sin y)j is conservative. If it is, find a potential function for it.

Is F(x, y) = 5x i + 5y j is a conservative vector field?

Determine whether is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)

Is F(x, y) = 2cos x i + 2cos y j is a conservative vector field?

Find where F(x, y) = 2(2xy2 + 1)i + 4x2y j and C is the curve r(t) = e t sin t i + e t cos t j; .

Let . Is the field conservative?

Determine whether is conservative. If it is, find a potential function for it. ( K is an arbitrary constant.)

Find the work done by the conservative force as it acts on a particle moving from P(0, 0) to .

Determine whether is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)

Determine whether F(x, y) = 5(2x + y3)i + 5(3xy2 - e-2y )j is conservative. If it is, find a potential function for it.

Is F(x, y) = 2y i + 2x j is a conservative vector field?

Is F(x, y) = 12cos y i + 12cos x j is a conservative vector field?

For F(x, y) = 3x2 i - 3y j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(3, 2) is

Determine whether F(x, y) = 6(y2 - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

The work done by F(x, y) =6 x i + 6xy j along the curve x = y² from (0, 0) to (1, 1) is

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin²t j; .

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y² - 3x sin y)j is conservative. If it is, find a potential function for it.

Find where F(x, y) = 2(2xy² + 1)i + 4x²y j and C is the curve r(t) = e ^t sin t i + e ^t cos t j; .

Determine whether F(x, y) = 5(2x + y³)i + 5(3xy² - e^-²^y )j is conservative. If it is, find a potential function for it.

For F(x, y) = 3x² i - 3y j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(3, 2) is

Determine whether F(x, y) = 6(y² - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.