Work

WORKDONE DUE TO VARIABLE FORCE: 

When the magnitude and direction of a force varies with position, the work done by such a force for an infinitesimal displacement is
given by  \( dW = \overrightarrow F .d\overrightarrow s \)     
          
               
The total work done in going from A to B as 
    shown in the figure is
        \( W = \int\limits_A^B {\overrightarrow F .d\overrightarrow s = \int\limits_A^B {\left( {F\cos \theta } \right)} } ds \)
In terms of rectangular component \( \overrightarrow F = F_x \hat i + F_y \hat j + F_z \hat k \)         
     \( d\overrightarrow s = dx\hat i + dy\hat j + dz\hat k \)
    \( \therefore W = \int\limits_A^B {\left( {F_x \hat i + F_y \hat j + F_z \hat k} \right)} .\left( {dx\hat i + dy\hat j + dz\hat k} \right) \)
    or \( W = \int\limits_{xA}^{xB} {F_x dx + } \int\limits_{yB}^{yB} {F_y dy} + \int\limits_{zA}^{zB} {F_z dz} \)