Gradient（梯度） of a scalar field

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本文介绍了标量场的梯度定义及其几何性质，通过实例解释了如何计算梯度，并探讨了梯度方向与变化率之间的关系。

Definition

The gradient of a scalar field $ϕ(x,y,z)\phi (x,y,z)$ is defined by
$\phi=\nabla \phi=\boldsymbol{i} \frac{\partial \phi}{\partial x}+\boldsymbol{j} \frac{\partial \phi}{\partial y}+\boldsymbol{k} \frac{\partial \phi}{\partial z}.$
Clearly, $∇ϕ\nabla \phi$ is a vector field whose $x$ -, $y$ - and $z$ - components are the first partial derivatives of $ϕ(x,y,z)\phi (x,y,z)$ with respect to $x$ , $y$ and $z$ respectively. Also note that the vector field $∇ϕ\nabla \phi$ should not be confused with the vector operator $ϕ∇\phi \nabla$ , which has components $(ϕ∂/∂x,ϕ∂/∂y,ϕ∂/∂z)(\phi\partial/\partial x, \phi\partial/\partial y, \phi\partial/\partial z)$ .

Examples

Example 1

Find the gradient of the scalar field $ϕ=xy2z3\phi=xy^{2}z^{3}$ .

Solution

From the definition, the gradient of $ϕ\phi$ is given by
$∇ϕ=i∂ϕ∂x+j∂ϕ∂y+k∂ϕ∂z=y2z3i+2xyz3j+3xy2z2k.\nabla \phi=\boldsymbol{i} \frac{\partial \phi}{\partial x}+\boldsymbol{j} \frac{\partial \phi}{\partial y}+\boldsymbol{k} \frac{\partial \phi}{\partial z}\\ =y^{2}z^{3}\boldsymbol{i}+2xyz^{3}\boldsymbol{j}+3xy^2z^2\boldsymbol{k}.$

Geometrical properties

The gradient of a scalar field $ϕ\phi$ has some interesting geometrical properties.
Let us first consider the problem of calculating the rate of change of $ϕ\phi$ in some particular direction. For an infinitesimal vector displacement $drd\boldsymbol{r}$ , forming its scalar product with $∇ϕ\nabla \phi$ we obtain
$∇ϕ⋅dr=(i∂ϕ∂x+j∂ϕ∂y+k∂ϕ∂z)⋅(idx+jdy+kdz)=∂ϕ∂xdx+∂ϕ∂ydy+∂ϕ∂zdz=dϕ,\nabla \phi \cdot d\boldsymbol{r}=(\boldsymbol{i} \frac{\partial \phi}{\partial x}+\boldsymbol{j} \frac{\partial \phi}{\partial y}+\boldsymbol{k} \frac{\partial \phi}{\partial z}) \cdot (\boldsymbol{i}dx+\boldsymbol{j}dy+\boldsymbol{k}dz)\\ =\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz\\ =d\phi,$
which is the infinitesimal change in $ϕ\phi$ in going from position $r\boldsymbol{r}$ to $r+dr\boldsymbol{r}+d\boldsymbol{r}$ . In particular, if $r\boldsymbol{r}$ depends on some parameter $u$ such that $r(u)\boldsymbol{r}(u)$ defines a space curve then the total derivative of $ϕ\phi$ with respect to $u$ along the curve is simply
$dϕdu=∇ϕ⋅drdu.\frac{d\phi}{du}=\nabla \phi \cdot \frac{d\boldsymbol{r}}{du}.$

In the particular case where the parameter $u$ is the arc length s along the curve, the total derivative of $ϕ\phi$ with respect to s along the curve is given by
$dϕds=∇ϕ⋅t^,\frac{d\phi}{ds}=\nabla \phi \cdot \hat{\boldsymbol{t}},$
where $t^\hat{\boldsymbol{t}}$ is the unit tangent to the curve at the given point.
In general, the rate of change of $ϕ\phi$ with respect to the distance s in a particular direction $a\boldsymbol{a}$ is given by
$dϕds=∇ϕ⋅a^\frac{d\phi}{ds}=\nabla \phi \cdot \hat{\boldsymbol{a}}$
and is called the directional derivative. Since $a^\hat{\boldsymbol{a}}$ is a unit vector we have
$dϕds=∣∇ϕ∣cosθ\frac{d\phi}{ds}=\lvert \nabla \phi \rvert cos\theta$
where $θ\theta$ is the angle between $a^\hat{\boldsymbol{a}}$ and $∇ϕ\nabla \phi$ as shown in figure below. Clearly $∇ϕ\nabla \phi$ lies in the direction of the fastest increase in $ϕ\phi$ , and $∣∇ϕ∣\lvert \nabla \phi \rvert$ is the largest possible value of $dϕ/dsd\phi/ds$ . Similarly, the largest rate of decrease of $ϕ\phi$ is $dϕ/ds=−∣∇ϕ∣d\phi/ds=-\lvert \nabla \phi \rvert$ in the direction of $−∇ϕ-\nabla \phi$ .
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