Mathematics: Find the work done by the force field f on a particle moving along the given path. f(x, y, z) = yzi + xzj + xyk c.line from (0, 0, 0) to (9, 7, 4)

Find the work done by the force field f on a particle

Ответ:

parkerpark9524

13.03.2022

Computing the line integral directly would be easy enough, but in general it's worth checking to see if the vector field is conservative; that is, whether there exists a scalar function

whose gradient corresponds exactly to the given vector field $\mathbf f$ , or $\nabla f=\mathbf f$ . Equivalently, we're looking for

such that

$\dfrac{\partial f}{\partial x}=yz$
$\dfrac{\partial f}{\partial y}=xz$
$\dfrac{\partial f}{\partial z}=xy$

We find that

$f_x=yz\implies f(x,y,z)=xyz+g(y,z)$
$f_y=xz+g_y=xz\implies g_y=0\implies g(y,z)=h(z)$
$f_z=xy+h_z=xy\implies h_z=0\implies h(z)=C$

and so there is indeed such a function

, with

.
Thus by the fundamental theorem of calculus, the line integral is path-independent, and

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{\mathcal C}\nabla f\cdot\mathrm d\mathbf r=f(\mathbf b)-f(\mathbf a)$

where $\mathcal C$ is any path starting at $\mathbf a$ and ending at $\mathbf b$ . Here,

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(9,7,4)-f(0,0,0)=252$

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Find the work done by the force field f on a particle moving along the given path. f(x, y, z) = yzi + xzj + xyk c.line from (0, 0, 0) to (9, 7, 4)

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