Mixing
Consider the vector field $F=-c frac{xmathbf{i}+ymathbf{j}}{x^2+y^2}$.
$begingroup$
$$mathbf{F}={-c}frac{xmathbf{i}+ymathbf{j}}{x^2+y^2}$$
(vector field was rewritten here to make it easier to see)
Consider the vector field above and using $c=1$, find by direct calculation the work done by the field in moving a unit mass along each of the following paths in the $xy$-plane.
a. $C$ is the half line $y=1$, $x geq 0$
b. C is the circle of radius a, with center at the origin, traced counterclockwise
c. C is the line from (0,1) to (1.0)
I actually have no idea where to start with this problem as i believe the professor never actually taught us this material but told us we would be assessed on it. I do believe that this is a line integral problem but I'm vexed as to how to set it up and solve it.
integration multivariable-calculus parametrization
edited Jan 31 at 1:15
jobe
1,109615
asked Jan 30 at 23:03
user63266user63266
92
$endgroup$
add a comment |
$begingroup$
$$mathbf{F}={-c}frac{xmathbf{i}+ymathbf{j}}{x^2+y^2}$$
(vector field was rewritten here to make it easier to see)
Consider the vector field above and using $c=1$, find by direct calculation the work done by the field in moving a unit mass along each of the following paths in the $xy$-plane.
a. $C$ is the half line $y=1$, $x geq 0$
b. C is the circle of radius a, with center at the origin, traced counterclockwise
c. C is the line from (0,1) to (1.0)
I actually have no idea where to start with this problem as i believe the professor never actually taught us this material but told us we would be assessed on it. I do believe that this is a line integral problem but I'm vexed as to how to set it up and solve it.
integration multivariable-calculus parametrization
edited Jan 31 at 1:15
jobe
1,109615
asked Jan 30 at 23:03
user63266user63266
92
$endgroup$
$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30
add a comment |
$begingroup$
$$mathbf{F}={-c}frac{xmathbf{i}+ymathbf{j}}{x^2+y^2}$$
(vector field was rewritten here to make it easier to see)
Consider the vector field above and using $c=1$, find by direct calculation the work done by the field in moving a unit mass along each of the following paths in the $xy$-plane.
a. $C$ is the half line $y=1$, $x geq 0$
b. C is the circle of radius a, with center at the origin, traced counterclockwise
c. C is the line from (0,1) to (1.0)
I actually have no idea where to start with this problem as i believe the professor never actually taught us this material but told us we would be assessed on it. I do believe that this is a line integral problem but I'm vexed as to how to set it up and solve it.
integration multivariable-calculus parametrization
edited Jan 31 at 1:15
jobe
1,109615
asked Jan 30 at 23:03
user63266user63266
92
$endgroup$
$$mathbf{F}={-c}frac{xmathbf{i}+ymathbf{j}}{x^2+y^2}$$
(vector field was rewritten here to make it easier to see)
Consider the vector field above and using $c=1$, find by direct calculation the work done by the field in moving a unit mass along each of the following paths in the $xy$-plane.
a. $C$ is the half line $y=1$, $x geq 0$
b. C is the circle of radius a, with center at the origin, traced counterclockwise
c. C is the line from (0,1) to (1.0)
I actually have no idea where to start with this problem as i believe the professor never actually taught us this material but told us we would be assessed on it. I do believe that this is a line integral problem but I'm vexed as to how to set it up and solve it.
integration multivariable-calculus parametrization
integration multivariable-calculus parametrization
edited Jan 31 at 1:15
jobe
1,109615
asked Jan 30 at 23:03
user63266user63266
92
edited Jan 31 at 1:15
jobe
1,109615
asked Jan 30 at 23:03
user63266user63266
92
edited Jan 31 at 1:15
jobe
1,109615
edited Jan 31 at 1:15
jobe
1,109615
edited Jan 31 at 1:15
jobe
1,109615
1,109615
asked Jan 30 at 23:03
user63266user63266
92
asked Jan 30 at 23:03
user63266user63266
92
asked Jan 30 at 23:03
user63266user63266
92
92
$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30
add a comment |
$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30
$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30
add a comment |
2 Answers
2
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oldest
votes
$begingroup$
For all three first you will need to parameterize the path.
a) $x = t, y = 1$
b) $x = acos t, y = asin t$
c) $x = 1-t, y =t$
For each scenario find $frac {dx}{dt}, frac {dy}{dt}$
And with the appropriate intervals of $t$ for each path.
$int_a^b left(frac {x(t)}{x(t)^2 + y(t)^2} frac{dx}{dt} + frac {y(t)}{x(t)^2 + y(t)^2} frac{dy}{dt}right) dt$
But, you might notice that $F$ is conservative (even though that is not being asked)
$F(x,y) = nabla left(-frac 12 c ln (x^2 + y^2)right)$
And so, for all of these, we can just check the endpoints.
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
$endgroup$
add a comment |
$begingroup$
I would first write the force vector as $frac{-cx}{x^2+y^2}vec{i}- frac{cy}{x^2+y^2}vec{j}$.
a) We can parameterize the "half line $xge 0$, y= 1" as x= t with t from 0 to $infty$, y= 1. $dx= dt$, dy= 0. So the integral $int_0^infty vec{F}cdot[dx, dy]= int_0^infty frac{-cx dx}{x^2+ 1}$. To integrate that, let $u= x^2+ 1$.
answered Jan 31 at 0:31
user247327user247327
11.5k1516
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
For all three first you will need to parameterize the path.
a) $x = t, y = 1$
b) $x = acos t, y = asin t$
c) $x = 1-t, y =t$
For each scenario find $frac {dx}{dt}, frac {dy}{dt}$
And with the appropriate intervals of $t$ for each path.
$int_a^b left(frac {x(t)}{x(t)^2 + y(t)^2} frac{dx}{dt} + frac {y(t)}{x(t)^2 + y(t)^2} frac{dy}{dt}right) dt$
But, you might notice that $F$ is conservative (even though that is not being asked)
$F(x,y) = nabla left(-frac 12 c ln (x^2 + y^2)right)$
And so, for all of these, we can just check the endpoints.
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
$endgroup$
add a comment |
$begingroup$
For all three first you will need to parameterize the path.
a) $x = t, y = 1$
b) $x = acos t, y = asin t$
c) $x = 1-t, y =t$
For each scenario find $frac {dx}{dt}, frac {dy}{dt}$
And with the appropriate intervals of $t$ for each path.
$int_a^b left(frac {x(t)}{x(t)^2 + y(t)^2} frac{dx}{dt} + frac {y(t)}{x(t)^2 + y(t)^2} frac{dy}{dt}right) dt$
But, you might notice that $F$ is conservative (even though that is not being asked)
$F(x,y) = nabla left(-frac 12 c ln (x^2 + y^2)right)$
And so, for all of these, we can just check the endpoints.
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
$endgroup$
add a comment |
$begingroup$
For all three first you will need to parameterize the path.
a) $x = t, y = 1$
b) $x = acos t, y = asin t$
c) $x = 1-t, y =t$
For each scenario find $frac {dx}{dt}, frac {dy}{dt}$
And with the appropriate intervals of $t$ for each path.
$int_a^b left(frac {x(t)}{x(t)^2 + y(t)^2} frac{dx}{dt} + frac {y(t)}{x(t)^2 + y(t)^2} frac{dy}{dt}right) dt$
But, you might notice that $F$ is conservative (even though that is not being asked)
$F(x,y) = nabla left(-frac 12 c ln (x^2 + y^2)right)$
And so, for all of these, we can just check the endpoints.
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
$endgroup$
For all three first you will need to parameterize the path.
a) $x = t, y = 1$
b) $x = acos t, y = asin t$
c) $x = 1-t, y =t$
For each scenario find $frac {dx}{dt}, frac {dy}{dt}$
And with the appropriate intervals of $t$ for each path.
$int_a^b left(frac {x(t)}{x(t)^2 + y(t)^2} frac{dx}{dt} + frac {y(t)}{x(t)^2 + y(t)^2} frac{dy}{dt}right) dt$
But, you might notice that $F$ is conservative (even though that is not being asked)
$F(x,y) = nabla left(-frac 12 c ln (x^2 + y^2)right)$
And so, for all of these, we can just check the endpoints.
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
answered Jan 31 at 0:43
Doug MDoug M
45.3k31954
45.3k31954
add a comment |
add a comment |
$begingroup$
I would first write the force vector as $frac{-cx}{x^2+y^2}vec{i}- frac{cy}{x^2+y^2}vec{j}$.
a) We can parameterize the "half line $xge 0$, y= 1" as x= t with t from 0 to $infty$, y= 1. $dx= dt$, dy= 0. So the integral $int_0^infty vec{F}cdot[dx, dy]= int_0^infty frac{-cx dx}{x^2+ 1}$. To integrate that, let $u= x^2+ 1$.
answered Jan 31 at 0:31
user247327user247327
11.5k1516
$endgroup$
add a comment |
$begingroup$
I would first write the force vector as $frac{-cx}{x^2+y^2}vec{i}- frac{cy}{x^2+y^2}vec{j}$.
a) We can parameterize the "half line $xge 0$, y= 1" as x= t with t from 0 to $infty$, y= 1. $dx= dt$, dy= 0. So the integral $int_0^infty vec{F}cdot[dx, dy]= int_0^infty frac{-cx dx}{x^2+ 1}$. To integrate that, let $u= x^2+ 1$.
answered Jan 31 at 0:31
user247327user247327
11.5k1516
$endgroup$
add a comment |
$begingroup$
I would first write the force vector as $frac{-cx}{x^2+y^2}vec{i}- frac{cy}{x^2+y^2}vec{j}$.
a) We can parameterize the "half line $xge 0$, y= 1" as x= t with t from 0 to $infty$, y= 1. $dx= dt$, dy= 0. So the integral $int_0^infty vec{F}cdot[dx, dy]= int_0^infty frac{-cx dx}{x^2+ 1}$. To integrate that, let $u= x^2+ 1$.
answered Jan 31 at 0:31
user247327user247327
11.5k1516
$endgroup$
I would first write the force vector as $frac{-cx}{x^2+y^2}vec{i}- frac{cy}{x^2+y^2}vec{j}$.
a) We can parameterize the "half line $xge 0$, y= 1" as x= t with t from 0 to $infty$, y= 1. $dx= dt$, dy= 0. So the integral $int_0^infty vec{F}cdot[dx, dy]= int_0^infty frac{-cx dx}{x^2+ 1}$. To integrate that, let $u= x^2+ 1$.
answered Jan 31 at 0:31
user247327user247327
11.5k1516
answered Jan 31 at 0:31
user247327user247327
11.5k1516
answered Jan 31 at 0:31
user247327user247327
11.5k1516
answered Jan 31 at 0:31
user247327user247327
11.5k1516
11.5k1516
add a comment |
add a comment |
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$begingroup$
any endpoints for scenario a). Do you how to calculate a path integral?
$endgroup$
– Doug M
Jan 30 at 23:55
$begingroup$
@DougM there are no endpoints for scenario a. If you're talking about a line integral, yes I know how to calculate that.
$endgroup$
– user63266
Jan 31 at 0:30