Is this Line Integral Independent of Path for a Conservative Field?

[unscientific]

Homework Statement

Evaluate this line integral ∫ F . dr , where F = (3x² sin y)i + (x³ cos y)j between the origin (0,0) and the point (2,4):

(a) along straight line y = 2x
(b) along curve y = x²

Homework Equations

The Attempt at a Solution

Part (a)
dr = dx i + dy j

∫ [ (3x² sin y) i + (x³ cos y)j ] . [dx i + dy j ]

= ∫ (3x² sin y)dx + (x³ cos y)dy

= ∫ d(x³ sin y) from [0,0] to [2,4]Does this mean that this line integral is independent of the path taken?

(b) If the line integral is independent of path, you should get the same answer..

Does dr = dx i + dy j still hold given that it's a curve now? do i have to use the "distance along curve" formula:

dr = √[ 1 + (dy/dx)² ] dxI've looked up RHB textbook it says it's fine to simply use dr = dx i + dy j ...

 

[Dick]

Homework Statement

Evaluate this line integral ∫ F . dr , where F = (3x² sin y)i + (x³ cos y)j between the origin (0,0) and the point (2,4):

(a) along straight line y = 2x
(b) along curve y = x²

Homework Equations

The Attempt at a Solution

Part (a)
dr = dx i + dy j

∫ [ (3x² sin y) i + (x³ cos y)j ] . [dx i + dy j ]

= ∫ (3x² sin y)dx + (x³ cos y)dy

= ∫ d(x³ sin y) from [0,0] to [2,4]


Does this mean that this line integral is independent of the path taken?

(b) If the line integral is independent of path, you should get the same answer..

Does dr = dx i + dy j still hold given that it's a curve now? do i have to use the "distance along curve" formula:

dr = √[ 1 + (dy/dx)² ] dx


I've looked up RHB textbook it says it's fine to simply use dr = dx i + dy j ...

x and y are not independent. You can't treat y as a constant when integrating dx and vice versa. Take your first path, y=2x. r=i dx+j dy=i dx+j 2*dx. Eliminate y from the integration by putting y=2x everywhere.

 

[unscientific]

x and y are not independent. You can't treat y as a constant when integrating dx and vice versa. Take your first path, y=2x. r=i dx+j dy=i dx+j 2*dx. Eliminate y from the integration by putting y=2x everywhere.

I know x and y are not independent, as they are bounded by y = 2x...

But my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x³ sin y)...

 

[Dick]

I know x and y are not independent, as they are bounded by y = 2x...

But my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x³ sin y)...

Yes it is.

 

[haruspex]

my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x³ sin y)...

You can think of F as a field resulting from the scalar potential x³ sin y. Since that is single-valued, the field must be conservative.