# Thread: calculation of div F, curl F, work

1. ## calculation of div F, curl F, work

Hello everybody

If anybody is able to help me with this problem (very long), I would greatly appreciate it. Since this is also an even numbered question, im not able to look up the answer. Special thanks to Scott who helped me with the Stoke's Theorem problem.

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Question:
Consider a force field given by: F(x,y,z) = y^2 i + 2 x y j + 2 z k

a) Calculate: div F

b) Calculate: curl F

c) Calculate the work required to move a particle along the parabolic path given by:
r(t) = t j + t^2 k , where -1 <= t <= 2

d) Calculate the work required to move a particle along a circular path of radius 10, centered at the origin, oriented counter-clockwise (relative to the positive z-axis) and with starting point (10,0).

Hello everybody

If anybody is able to help me with this problem (very long), I would greatly appreciate it. Since this is also an even numbered question, im not able to look up the answer. Special thanks to Scott who helped me with the Stoke's Theorem problem.

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Question:
Consider a force field given by: F(x,y,z) = y^2 i + 2 x y j + 2 z k

a) Calculate: div F

b) Calculate: curl F

c) Calculate the work required to move a particle along the parabolic path given by:
r(t) = t j + t^2 k , where -1 <= t <= 2

d) Calculate the work required to move a particle along a circular path of radius 10, centered at the origin, oriented counter-clockwise (relative to the positive z-axis) and with starting point (10,0).
a) and b) are nothing more than aplication of the definition and calculating partial derivatives. Where are you stuck?

c) $\displaystyle W = \int_C F \cdot dr = \int_C F \cdot \frac{dr}{dt} \, dt$

Note that on the curve, $\displaystyle x = 0, ~ y = t, ~ z = t^2$. Therefore:

$\displaystyle W = \int_{t = -1}^{t = 2} ((t^2)^2 i + 0 j + 2(t^2) k) \cdot (j + 2t k) \, dt = ....$

d) Start by getting the parametric equations of the path. Then it's similar to c).

Note that if $\displaystyle \nabla \times F = 0$ (and I haven't checked to see whether it does or not) then $\displaystyle F = \nabla \phi$ - find $\displaystyle \phi$ and c) and d) become much simpler to do ....

3. ## thank mr. fantastic

for parts a and b, i really dont understand the definition, I am confused on how to apply it For part d, I don't know how to get the parametric equation

Thanks Mr. Fantastic, you really are fantastic

for parts a and b, i really dont understand the definition, I am confused on how to apply it For part d, I don't know how to get the parametric equation

Thanks Mr. Fantastic, you really are fantastic
What part of the definitions don't you understand?

d) $\displaystyle x = 10 \cos t$ and $\displaystyle y = 10 \sin t$ and the starting point is $\displaystyle t = 0$.

5. ## how to apply definition to problem

The divergence of F ( x, y, z ) = M i + N j + P k is

div F ( x, y, z ) = (upside down triangle) * F ( x, y, z ) = dM/dx + dN/dy + dP/dz

If div F = 0 , then F is said to be divergent free.

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I don't know how to apply this definition to the problem?? Is this definition even correct? Like do I just take the derivative of the function? Thanks for taking the time to help me mr. fantastic

The divergence of F ( x, y, z ) = M i + N j + P k is

div F ( x, y, z ) = (upside down triangle) * F ( x, y, z ) = dM/dx + dN/dy + dP/dz

If div F = 0 , then F is said to be divergent free.

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I don't know how to apply this definition to the problem?? Is this definition even correct? Like do I just take the derivative of the function? Thanks for taking the time to help me mr. fantastic
To get div F, note that M = y^2, N = 2xy and P = 2z. Now take the required partial derivatives and substitue them into the definition.

7. ## u rock!!!

Originally Posted by mr fantastic
To get div F, note that M = y^2, N = 2xy and P = 2z. Now take the required partial derivatives and substitue them into the definition.
you rock fantastic!!!