3. The Gradient of a Function

Directional Derivative

Theorem

For a differentiable function $f(x,y)$, a unit vector $u=⟨u_1,u_2⟩$ and a point $(a,b)$ in the domain of $f$, we have

$D_{u}f(a,b)=||\nabla f(a,b)||\cos (\theta )$

where $\theta$ is the angle between the vector $u$ and the gradient vector.

Proof: The dot product of two vectors v and w is always given by

$v\cdot w=||v||||w||\cos (\theta )$

where $\theta$ is the angle between v and w.

Steepest Ascent

Theorem

Let $f(x,y)$ be a differentiable function. Then the gradient vector $v=\nabla f(x,y)$ indicates the direction of steepest ascent and, if $\hat{v}$ denotes the corresponding normalized vector, we have

$D_{\hat{v}}f(x,y)=||\nabla f(x,y)||$

Steepest Descent

Theorem

Let $f(x,y)$ be a differentiable function. Then the vector $v=-\nabla f(x,y)$ indicates the direction of steepest descent and, if $\hat{v}$ denotes the corresponding normalized vector, we have

$D_{\hat{v}}f(x,y)=-||\nabla f(x,y)||$

Directional Derivative

Theorem

For a differentiable function $f(x,y)$, a unit vector $u=⟨u_1,u_2⟩$ and a point $(a,b)$ in the domain of $f$, we have

$D_{u}f(a,b)=||\nabla f(a,b)||\cos (\theta )$

where $\theta$ is the angle between the vector u and the gradient vector.

Gradient VS Level Curve

Theorem

For a differentiable function $f(x,y)$, the gradient vector $\nabla f(a,b)$ is perpendicular to the tangent line to the level curve $f(x,y)=f(a,b)$ at the point $(a,b)$.

Tangent Line to Level Curve

Theorem

Suppose $f(x,y)$ is a differentiable function and $k$ is an arbitrary constant, then the tangent line to the curve defined by $f(x,y)=k$ at the point $(a,b)$ is

$f_x(a,b)x+f_y(a,b)y=c$

where $c=f_x(a,b)+f_y(a,b)b$.

Tangent Plane to Level Surface

Theorem

Suppose $f(x,y,z)$ is a differentiable function and $k$ is an arbitrary constant. Then the tangent plane to the surface defined by $f(x,y,z)=k$ at the point $(a,b,c)$ is given by the equation

$f_x(a,b,c)x+f_y(a,b,c)y+f_z(a,b,c)z=C$

where $C=f_x(a,b,c)a+f_y(a,b,c)b+f_z(a,b,c)c$.