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3.4 The Gradient

3.4.1 Definition of the Gradient

Definition: Let f(x,y) be a real function of two variables. The gradient gradf of f in the point ({x}_{0},{y}_{0}) in the x–y–plane is the two–component vector of the partial derivatives {f}_{x} and {f}_{y} of f,

\begin{eqnarray} gradf({x}_{0},{y}_{0}) ≡\mathop{∇}f({x}_{0},{y}_{0}) = \left ({f}_{x}({x}_{0},{y}_{0}),{f}_{y}({x}_{0},{y}_{0})\right ).& & %&(3.14) \\ \end{eqnarray}

The symbol \mathop{∇} is called ‘Nabla’–operator. Note: the gradient of f in the point ({x}_{0},{y}_{0}) is a two–dimensional vector in the x–y–plane attached to that point. The map (x,y) →\mathop{∇}f(x,y) defines a vector field, i.e. to each point (vector) (x,y) in the x–y–plane, a vector \mathop{∇}f(x,y) is attached.

3.4.2 Examples

Paraboloid f(x,y) = {x}^{2} + {y}^{2}

In this case,

\begin{eqnarray*} \mathop{∇}f(x,y) = (2x,2y).& & %& \\ \end{eqnarray*}

Sketch this vector field in the x–y–plane (solution is given in the lecture).

Hyperboloid f(x,y) = {x}^{2} − {y}^{2}

In this case,

\begin{eqnarray*} \mathop{∇}f(x,y) = (2x,−2y).& & %& \\ \end{eqnarray*}

Sketch this vector field in the x–y–plane (solution is given in the lecture).

3.4.3 Gradient and Differential; Geometrical Meaning

Reminder: Total change of height (total differential)

\begin{eqnarray*} df(x,y) := {∂\over ∂x}f(x,y)dx + {∂\over ∂y}f(x,y)dy& & %& \\ \end{eqnarray*}

is called the total differential of f(x,y) at the point (x,y) and gives the total change of the height of the surface f(x,y) (measured from the x–y–plane) at the point (x,y), if we move a tiny step dx along the x–direction and a tiny step dy along the y–direction.

Consider now a certain point ({x}_{0},{y}_{0}) in the x–y–plane, with the gradient \mathop{∇}f({x}_{0},{y}_{0}) of the function f(x,y) attached. In that point, the total change of height of the function f(x,y) can be written as a scalar product,

\begin{eqnarray*} df = {∂\over ∂x}fdx + {∂\over ∂y}fdy = ({f}_{x},{f}_{y}) ⋅ (dx,dy)& & %& \\ \end{eqnarray*}

of the two vectors ({f}_{x},{f}_{y}) = \mathop{∇}f(≡\mathop{∇}f({x}_{0},{y}_{0})) and (dx,dy). We now change dx and dy slightly, thereby changing the vector (dx,dy) of the differentials. Then, for a certain values of dx and dy, the vector (dx,dy) becomes perpendicular to the gradient ({f}_{x},{f}_{y}), i.e. the scalar product df = \mathop{∇}f ⋅ (dx,dy) vanishes. In this direction (dx,dy), the height of the surface does not change, it determines the direction of an equipotential line. Therefore, the gradient \mathop{∇}f({x}_{0},{y}_{0}) is perpendicular to the equipotential line through ({x}_{0},{y}_{0}); it determines the direction of the steepest increase of the function f(x,y).

Example: Paraboloid f(x,y) = {x}^{2} + {y}^{2}

We have

\begin{eqnarray*} \mathop{∇}f(x,y) = (2x,2y).& & %& \\ \end{eqnarray*}

The equipotential lines are circles {r}^{2} = {x}^{2} + {y}^{2} in the x–y–plane. The gradient is perpendicular to these circles. Picture in the lecture.

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