geometrical meaning of partial derivatives

Question

If $f$ is a function of $x,y,z$, then what is the geometrical mean of $\frac {\partial f}{\partial x}$ ? Similary what is geometrical meaning of other partial derivatives of function $f$. Please tell ... Thanks

Helpful Source: https://www.math.ucdavis.edu/~zekius/s13mat21c/handouts/Partial.pdf — Saharsh, Mar 07 '15 at 13:59

score 1 · Answer 1 · answered Mar 07 '15 at 16:27

Well if it is difficult to comprehend the meaning of partial derivatives in the hyperspace $\mathbb{R}^4 $ then we can first talk about the simpler and easily comprehensible one $\mathbb{R}^3$- space. When we try to find the derivative of $z=f(x, y)$ at $(x_0, y_0)$ with respect to, say $x$, then we consider $y$ to be a constant i. e. we actually consider the intersection of the surface $f(x, y)$ and the plane $y = y_0$ which gives us a curve in $y = y_0$ plane (an $\mathbb{R}^2$-space). Then as it is obvious that $\frac{\partial z} {\partial x} $ gives us the slope of tangent at the point $(x_0,y_0)$ with respect to $x$-axis.

We may now go ahead to generalize this concept for hyperspace by saying that the intersection of the hyperplane $z=z_0$ and the hypersurface $w = f(x, y, z)$ is a surface in $\mathbb{R}^3 $ and the same story repeats which has been told earlier but in the end the fact remains the same that $\frac{\partial z} {\partial x} $ denotes slope.

score 0 · Answer 2 · answered Mar 07 '15 at 13:45

0

You can think of $\frac {\partial f}{\partial x}$ as the slope in the x-direction.

answered Mar 07 '15 at 13:45

BadAtMaths

307

1

I would say 'the slope with regard to the x axis'. – Mar 07 '15 at 13:46
1

The slope of what? The graph $w=f(x,y,z)$ lives in $\mathbb{R}^4$, so it might not be very easy to visualize its slope... – Hans Lundmark Mar 07 '15 at 14:14

score 0 · Answer 3 · answered Mar 07 '15 at 16:01

From a geometrical point of view:

If $f$ is smooth enough, then

$d f= \frac{\partial f}{\partial x}d x+ \frac{\partial f}{\partial y}d y+ \frac{\partial f}{\partial z}d z$

is the local representation of the total differential of $f$.

It's an covector and lives in the dual of $R^3$.

The $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$ are just local coordinates of this covector.

geometrical meaning of partial derivatives

3 Answers3