The gravitational field in a region is given by $$\overrightarrow g = \frac{{5N}}{{kg\hat i}} + \frac{{12N}}{{kg\hat j}}.$$    The change in the gravitational potential energy of a particle of mass $$1\,kg$$  when it is taken from the origin to a point $$\left( {7\,m, - 3\,m} \right)$$   is :

Solution :
$$\eqalign{ & {\text{Gravitational field,}}\,\,I = \left( {5\hat i + 12\hat j} \right)N/kg \cr & I = - \frac{{dv}}{{dr}} \cr & v = - \left[ {\int\limits_0^x {{I_x}} dx + \int\limits_0^y {{I_y}} dy} \right] = - \left[ {{I_x}.x + {I_y}.y} \right] \cr & = - \left[ {5\left( {7 - 0} \right) + 12\left( { - 3 - 0} \right)} \right] \cr & = - \left[ {35 + \left( { - 36} \right)} \right] = 1\,J/kg \cr} $$
i.e., change in gravitational potential $$1\,J/kg.$$  Hence change in gravitational potential energy $$1\,J.$$