The gradient of a scalar field v is a vector that represents both the magnitude and direction of the maximum space rate of increase of v. obtain a mathematical expression for the gradient of v using calculus by evaluating the difference in field dv between two points on contours where v is a constant. show that the maximum rate of increase of v with respect to occurs when dil is the normal derivative dn.

Global warming will produce rising sea levels partly due to melting ice caps but also due to the expansion of water as average ocean temperatures rise. to get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of 1.00ºc. note that this calculation is only approximate because ocean warming is not uniform with depth. (answer in ×10^{-3} −3 m)