Question

1. Consider the force field \textbf{F}(x, y, z) = (2xyz^2, x^2z^2 + z \cos yz, 2x^2yz + y \cos yz). Show that the integral $\int_C \textbf{F} \cdot d\textbf{r}$ is independent of the path. Then use the fundamental theorem of line integrals to calculate the work done by \textbf{F} on an object moving along a curve C from (0, 0, 1) to $(1, \frac{\pi}{4}, 2)$.

          1.
Consider the force field \textbf{F}(x, y, z) = (2xyz^2, x^2z^2 + z \cos yz, 2x^2yz + y \cos yz). Show that the integral $\int_C \textbf{F} \cdot d\textbf{r}$ is independent of the path. Then use the fundamental theorem of line integrals to calculate the work done by \textbf{F} on an object moving along a curve C from (0, 0, 1) to $(1, \frac{\pi}{4}, 2)$.

1.
Consider the force field F(x, y, z) = (2xyz^2, x^2z^2 + z cosyz, 2x^2yz + y cosyz). Show that the integral F· dr is independent of the path. Then use the fundamental theorem of line integrals to calculate the work done by F on an object moving along a curve C from (0, 0, 1) to (1, (π)/(4), 2).

Added by Lu-S J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

08/18/2022

Step 1

A force field is conservative if it has a potential function, which means that the curl of F is zero. Mathematically, this can be expressed as: curl(F) = (∂/∂y(F_z) - ∂/∂z(F_y), ∂/∂z(F_x) - ∂/∂x(F_z), ∂/∂x(F_y) - ∂/∂y(F_x)) = (0, 0, 0) Show more…

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Consider the force field F(x,y,z)=(:2xyz^(2),x^(2)z^(2)+zcosyz,2x^(2)yz+ycosyz:). Show that the integral int_C F*dr. is independent of the path. Then use the fundamental theorem of line integrals to calculate the work done by F on an object moving along a curve C from (0,0,1) to (1,(pi )/(4),2) 1. Consider the force field F(x,y,z) =(2xyz2,x2z2+z cos yz,2x2yz+ycos yz). Show that the integral F - dr. is independent of the path. Then use the fundamental theorem (0,0,1) to (1,7,2)