What does the gradient of a vector field give us?

I know that a vector field is a space in which every point is associated with a vector. But what would I be meaning if I talk about gradient of that vector field? What does this gradient actually signify?|||the gradient is a vector; you find the gradient by taking directional derivatives of a scalar field; the gradient tells you the rate of change of the quantity of interest





for instance, suppose you measure the temperature of a room with hundreds of thermometers and can plot the gradient of the temperature field...where the gradient is largest is where the temperature is changing most rapidly, where the gradient is near zero is where the temperature is constant|||Or, another example, check out the weather maps on TV. They show lines across the surface. These lines are called isobars, lines where atmospheric pressure is the same.





When the isobars are closer together, the gradient is steeper than when the isobars are farther apart. The gradient is steeper where the lines are closer because the pressure change per distance dP/dS is greater. In practice the dP = constant on weather maps; so the shrinking distance between them ds %26lt; dS makes the gradient steeper.





In this example, the field is a pressure field, the atmospheric pressure. And at each point along the surface of the Earth, there is a tiny little pressure vector pushing onto the surface. [See source.]|||the slope of the vector field at that point?