To find the flow of F over the curve r(t), we need to compute the line integral of F over the curve r(t).

Step 1: Parameterize the curve r(t)
The curve r(t) is already parameterized as r(t) = t^0.5 i + t j.

Step 2: Compute dr/dt
The derivative of r(t) with respect to t is dr/dt = 0.5t^-0.5 i + j.

Step 3: Substitute r(t) into F
Substitute r(t) into F to get F(r(t)) = 0.25(t^0.5*t i + (t-t^0.5) j).

Step 4: Compute F(r(t)) . dr/dt
The dot product of F(r(t)) and dr/dt is F(r(t)) . dr/dt = 0.25[0.5t^0.5*t*t^-0.5 + (t-t^0.5)].

Step 5: Integrate F(r(t)) . dr/dt from 0 to 4
The flow of F over the curve r(t) is the integral of F(r(t)) . dr/dt from 0 to 4.

This integral can be computed using standard techniques of calculus. The exact value will depend on the specific form of the function F and the curve r(t).

Question

To find the flow of F over the curve r(t), we need to compute the line integral of F over the curve r(t).

Step 1: Parameterize the curve r(t)
The curve r(t) is already parameterized as r(t) = t^0.5 i + t j.

Step 2: Compute dr/dt
The derivative of r(t) with respect to t is dr/dt = 0.5t^-0.5 i + j.

Step 3: Substitute r(t) into F
Substitute r(t) into F to get F(r(t)) = 0.25(t^0.5*t i + (t-t^0.5) j).

Step 4: Compute F(r(t)) . dr/dt
The dot product of F(r(t)) and dr/dt is F(r(t)) . dr/dt = 0.25[0.5t^0.5*t*t^-0.5 + (t-t^0.5)].

Step 5: Integrate F(r(t)) . dr/dt from 0 to 4
The flow of F over the curve r(t) is the integral of F(r(t)) . dr/dt from 0 to 4.

This integral can be computed using standard techniques of calculus. The exact value will depend on the specific form of the function F and the curve r(t).

Knowee AI · Accepted Answer

To find the flow of F over the curve r(t), we need to compute the line integral of F over the curve r(t).

Step 1: Parameterize the curve r(t)
The curve r(t) is already parameterized as r(t) = t^0.5 i + t j.

Step 2: Compute dr/dt
The derivative of r(t) with respect to t is dr/dt = 0.5t^-0.5 i + j.

Step 3: Substitute r(t) into F
Substitute r(t) into F to get F(r(t)) = 0.25(t^0.5*t i + (t-t^0.5) j).

Step 4: Compute F(r(t)) . dr/dt
The dot product of F(r(t)) and dr/dt is F(r(t)) . dr/dt = 0.25[0.5t^0.5*t*t^-0.5 + (t-t^0.5)].

Step 5: Integrate F(r(t)) . dr/dt from 0 to 4
The flow of F over the curve r(t) is the integral of F(r(t)) . dr/dt from 0 to 4.

This integral can be computed using standard techniques of calculus. The exact value will depend on the specific form of the function F and the curve r(t).

Salt water with density of 0.25 g/cm2 flows over the curve r(t) = t0.5 i + t j, 0 ≤ t ≤ 4,according to the vector field F = 0.25v where v = xy i + (y-x) j is the velocity fieldmeasured in cm/s. Find the flow of F over the curve r(t)

Question

Solution

Similar Questions

Upgrade your grade with Knowee