Significance of the Gradient Vector

A surface defined by a function of two variables can be understood by examining how it changes along specific directions. When one variable is held constant, the surface reduces to a curve that reflects variation in the other variable. For example, fixing one variable and moving parallel to a coordinate axis produces a cross-sectional curve. The slope of this curve at a given point represents how the function changes in that particular direction, providing a measure of local steepness.

By analyzing these cross-sectional curves along each axis, one obtains directional rates of change. Each of these describes how the surface varies when movement is restricted to a single coordinate direction. While these directional measures are useful, they only capture changes along the axes and do not fully describe how the surface behaves in all possible directions.

The concept of the gradient extends this idea by combining the directional changes into a single vector quantity. This vector points in the direction where the function increases most rapidly at a given point. Its magnitude indicates how steep the surface is in that direction. A larger magnitude corresponds to a sharper incline, whereas a smaller magnitude indicates a gentler slope.

In practical applications such as civil engineering, the gradient has a clear physical meaning. It represents the incline of a road or surface. A steeper gradient increases the effort required for vehicles to move upward and often reduces their speed. In contrast, a gentler gradient allows for smoother and more efficient movement, requiring less energy to overcome resistance.