Definition: Let be a real function of two variables. The gradient of in the point in the – –plane is the two–component vector of the partial derivatives and of ,
The symbol is called ‘Nabla’–operator. Note: the gradient of in the point is a two–dimensional vector in the – –plane attached to that point. The map defines a vector field, i.e. to each point (vector) in the – –plane, a vector is attached.
In this case,
Sketch this vector field in the – –plane (solution is given in the lecture).
In this case,
Sketch this vector field in the – –plane (solution is given in the lecture).
Reminder: Total change of height (total differential)
is called the total differential of at the point and gives the total change of the height of the surface (measured from the – –plane) at the point , if we move a tiny step along the –direction and a tiny step along the –direction.
Consider now a certain point in the – –plane, with the gradient of the function attached. In that point, the total change of height of the function can be written as a scalar product,
of the two vectors and . We now change and slightly, thereby changing the vector of the differentials. Then, for a certain values of and , the vector becomes perpendicular to the gradient , i.e. the scalar product vanishes. In this direction , the height of the surface does not change, it determines the direction of an equipotential line. Therefore, the gradient is perpendicular to the equipotential line through ; it determines the direction of the steepest increase of the function .
We have
The equipotential lines are circles in the – –plane. The gradient is perpendicular to these circles. Picture in the lecture.