Consider a function $g: \mathbb{R^3} \to \mathbb{R}$ defined by $g(x,y,z) = z^2 -x^3 + 2z + 3y^3$
Find the gradient of $g$ at the point $(2,1,-1)$.
Is the gradient $\nabla g(2,1,-1)$ given by a vector, that is, $\nabla g(2,1,-1) = -12i + 9j$? If so, then what does the directional derivative mean?
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$\begingroup$The gradient is a vector; it points in the direction of steepest ascent.
The directional derivative is a number; it is the rate of change when your point in $\Bbb R^3$ moves in that direction. (You can imagine "reducing" your function to a function of a single variable, say $t$, by "slicing" the curve in that direction; the directional derivative is just then the 1-D derivative of that "sliced" function.)
$\endgroup$ 1 $\begingroup$Be careful that directional derivative of a function is a scalar while gradient is a vector.
The only difference between derivative and directional derivative is the definition of those terms. Remember:
- Directional derivative is the instantaneous rate of change (which is a scalar) of $f(x,y)$ in the direction of the unit vector $u$.
- Derivative is the rate of change of $f(x,y)$, which can be thought of the slope of the function at a point $(x_0,y_0)$.
Yes, the gradient is given by the row vector whose elements are the partial derivatives of $g$ with respect to $x$, $y$, and $z$, respectively. In your case the gradient at $(x,y,z)$ is hence $[-3x^2,9y^2,2z+2]$. The gradient at $(2,1,-1)$ is therefore $[-12,9,0]$.
The directional derivative at a point $(x,y,z)$ in direction $(u,v,w)$ is the gradient multiplied by the direction divided by its length. So if $u^2+v^2+w^2=1$ then the directional derivative at $(x,y,z)$ in direction $(u,v,w)$ is just $-3x^2 u + 9y^2v + (2z+2)w$.
If $u^2+v^2+w^2\neq 1$ then you should divide the number above by $\sqrt{u^2+v^2+w^2}$.
In sum, the gradient is a vector with the slope of the function along each of the coordinate axes whereas the directional derivative is the slope in an arbitrary specified direction.
$\endgroup$ $\begingroup$- A Directional Derivative is a value which represents a rate of change
- A Gradient is an angle/vector which points to the direction of the steepest ascent of a curve.
Let us take a look at the plot of the following function:
$$ \bbox[lightgray] {f(x) = -x^2+4}\qquad (1)$$
The 1st derivative of the function is:
$$ \bbox[lightgray] {\frac{dy}{dx} = -2x}\qquad (2)$$
Putting $x=-1$ in $(2)$ we obtain,
$\implies \frac{dy}{dx} = \frac{rise}{run}= 2 ... ... ...\qquad (3)$
Also,
$tan \theta = 2$
$\implies \theta = tan^{-1}(2)$
$\implies \theta = 0.964 $ radian
So, $\theta = 55.23 ^\circ ... ... ...\qquad (4)$
.
Similarly, putting $x=-2$ in $(2)$ we obtain,
$\implies \frac{dy}{dx} = \frac{rise}{run}= 4 ... ... ...\qquad (5)$
So, $\theta = 57.25 ^\circ ... ... ...\qquad (6)$
So, the answer is:
- (3) and (5) are Directional Derivatives.
- Directional Derivatives are scalar values.
And,
- (4) and (6) are Gradients.
- Gradients are vector values.
Taking a function $g: \mathbb{R}^3\rightarrow \mathbb{R}$. Its gradient $\nabla{g}$ is a vector, while the directional derivative is the inner product of another vector $\mathbf{v}$ in the same space with $\nabla g$, denoted as $\mathbf{v}\cdot\nabla g$. It means the projection of the vector $\nabla g$ onto the direction of the vector $\mathbf{v}$, which represents how the function $g$ varies in that direction.
$\endgroup$ $\begingroup$In simple words, directional derivative can be visualized as slope of the function at the given point along a particular direction. For example partial derivative w.r.t x of a function can also be written as directional derivative of that function along x direction.
Gradient is a vector and for a given direction, directional derivative can be written as projection of gradient along that direction.
Please go through this link to have a clear idea:
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