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Say that:

$$z = xy$$


$${\partial z \over \partial x} = y$$


$${\partial z \over \partial y} = x$$

If we plot in 3D space the 2D surface corresponding to eq1, than take a point on that surface, the tangent with respect to the x axis is y, and the tangent corresponding to the y axis is x.

Do the total derivatives ($dz \over dx$ and $dz \over dy$) have a similar geometric interpretation?

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Total derivative is $dz=z_xdx+z_yd_y$. –  Nancy Rutkowskie Oct 26 '12 at 20:25
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3 Answers

Here are two ways to think about the total derivative of a function $z$.

  1. If the gradient vector field is $$\nabla z = \begin{pmatrix}\frac{\partial z}{\partial x} \\ \frac{\partial z}{\partial y}\end{pmatrix},$$ the total derivative is a "covector field" given by the transpose of the gradient, $$dz = \begin{pmatrix}\frac{\partial z}{\partial x}\ \ \frac{\partial z}{\partial y}\end{pmatrix}.$$ What information does this encode? Directional derivatives. If $v = \begin{pmatrix}v_1 \\ v_2\end{pmatrix}$ is a vector field, then matrix-multiplying $(dz)(v)$ is the function that tells you the directional derivative of $dz$ along $v$ at each point.

  2. Remember from calculus that the derivative is the slope of the best affine-linear approximation to a function. In other words, if you zoom in close to $x_0$, then $f$ begins to look very much like $f(x_0) + f'(x_0)(x-x_0)$. In this context, $dz$ is the "best linear approximation" to $z$ at $(x,y)$. If you zoom in close to $(x_0,y_0)$, the map $z$ looks very much like the affine-linear map $$x_0y_0 + \begin{pmatrix}\frac{\partial z}{\partial x} \ \ \frac{\partial z}{\partial y}\end{pmatrix}\begin{pmatrix}x-x_0 \\ y-y_0\end{pmatrix}.$$ The first term is $z(x_0,y_0) = x_0y_0$, and again we are matrix-multiplying in the second term.

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Expanding slightly on my comment on your other question: the total derivative is not taken in a specific direction (i.e. with respect to a particular variable). Rather, it is roughly a vector whose components are given by the partial derivatives. If we specialize to a point, the each partial derivative gives the slope of a tangent line in the given direction. At a smooth point, these lines are distinct, i.e. linearly independent, so they determine a plane. Just as you can imagine a smoothly varying tangent line moving with a point on a curve, you can imagine a smoothly varying tangent plane moving with a point on a surface.

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The total derivative may be used in dynamic geometry. See Geogebra. The addition of time to geometry was a part of Newton's calculus. The static $(x,y,z)$ coordinate system expanded to $(x,y,z,t)$. The static coordinates expanded to $[x(t),y(t),z(t)]$. One derivative of interest is $dz/dt$. The independent variable is time which is plotted along the $x$-axis. The dependent variables $(x,y,z)$ are plotted individually along the vertical axis. Also, in geometry $z$ may be plotted as the radius of an expanding circle $x(t)^2 + y(t)^2 = z(t)^2$. This adds time to the Pythagorean theorem. In special relativity $z=ct$ where $c$ is the speed of light. This geometry is the light cone. Newton was interested in the orbits of planets. This requires keeping track of the directions of motions with vectors. Fortunately the motion of the earth is mostly in a plane. In this case the object in plane geometry is the ellipse for closed orbits. It is a hyperbola or parabola for open orbits. Another geometric item of interest is the curvature. For a circle it is $1/r$ and for a sphere it is $1/r^2$. The $1/r^2$ is interpreted as the gravitational force. Euler proved that the second derivative with respect to time is also the curvature. This second derivative is interpreted as the acceleration. Thus Newton's second law states that temporal curvature must be equal to spatial curvature using local masses and a universal constant as parameters. This was the dynamic geometry that was used before Einstein created his geometry of gravity, known as General Relativity. Quantum mechanics required the development of a completely different type of geometry.

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Consider separating text into paragraphs for ease of reading. –  Epictetus Dec 14 '12 at 9:11
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