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Say I have variables $x,y_1,y_2,z_1,z_2$ all $\in \mathbb{R}$

And I have the following equations:

$$x = f_1(y_1,y_2)$$ $$y_1 = f_2(z_1,z_2)$$

How does:

$$dx \over dz_1$$

differ from:

$$\partial x \over \partial z_1$$

or am I confused?

Intuitively I just want to think about how $x$ varies in proportion to an infinitesimally small perturbation of $z_1$, so I don't understand the difference between the two different notations (nonpartial vs partial)?

  • 1
    As a note, I would avoid the use of the word "normal" in your title. "Normal derivative" might be interpreted by some to mean the gradient in the normal direction!2012-10-26
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    Let's say you had $x=z_1z_2$. Then $\frac{\partial x}{\partial z_1} = z_2$ but $\frac{dx}{dz_1} = z_2 + z_1\frac{dz_2}{dz_1}$. (I haven't put this as an answer because you probably need more explanation than this.)2012-10-26
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    Now I am really confused. If we plot in 3D a surface $x=z_1z_2$, then at a given point on the surface the tangent in the $z_1$ direction will correspond to the partial derivative $\partial x \over \partial z_1$. Does the total derivative have a similiar geometric interpretation?2012-10-26
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    http://math.stackexchange.com/questions/221755/geometric-interpretation-of-total-derivative2012-10-26
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    The total derivative corresponds to the tangent plane to the surface at the given point; of course this plane is spanned by the tangent lines corresponding to $\partial x/\partial z_1$ and $\partial x/\partial x_2$. In higher dimensions the total derivative corresponds to the tangent space (whose dimension $n$ is the same as that of the given manifold), which is spanned by the $n$ first partials.2012-10-26

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