Please don't laugh at the following: I am new to multivariable calculus, and just thought of this on the fly:(adsbygoogle = window.adsbygoogle || []).push({});

In single variable calculus, for a function [itex] y = f(x)[/itex] we have the tangent line approximation:

[tex] \Delta y \approx \frac{df}{dx} \Delta x [/tex]

The relation becomes exact in the limit:

[tex] dy = \frac{df}{dx} dx [/tex]

So to find the total change in [itex]y [/itex] over an interval [itex] [a,b] [/itex] we just have:

[tex] \int dy = \int_{a}^{b} \frac{df}{dx} dx = \int_{a}^{b} f^{\prime} (x)dx [/tex]

I hope that this reasoning is correct. Having assumed that it was, and being verynaive, I tried to extend this result to the multivariable case using the tangent plane approximation. For a function [itex] z = f(x, y)[/tex]

[tex] \Delta z \approx \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y [/tex]

In the limit:

[tex] dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy [/tex]

So I just assumed that for a change in [itex] x [/itex] from [itex] a [/itex] to [itex] b [/itex], and a change in [itex] y [/itex] from [itex] c [/itex] to [itex] d [/itex], the total change in [itex] z [/itex] was given by:

[tex] \int_{a}^{b} \frac{\partial f}{\partial x} dx + \int_{c}^{d} \frac{\partial f}{\partial y} dy [/tex]

[tex] = \int_{a}^{b} f_{x} (x,y) dx + \int_{c}^{d} f_{y} (x,y) dy [/tex]

Now, I know that this isdead wrong. I can think of a couple of reasons. For one thing, the curly d’s set off alarm bells: if [itex] f_{x} [/itex] has [itex] y [/itex] in it, then the first integral cannot be evaluated, and if [itex] f_{y} [/itex] has [itex] x [/itex] in it, then the second integral cannot be evaluated. Also, the region over which [itex] z [/itex] is changing is limited to a rectangle. My question is: where did I go wrong in this line of reasoning? I have a vague inkling that the problem is that you cannot do it in two parts using the partials…you need something else to be the “whole” derivative of the multivariable function. My text book says that the net change in a function [itex] z = f(x, y)[/tex] can be evaluated using a double integral, but I’m not sure how they factor in. To me, [itex] \iint [/itex] seems to be something else entirely. I cannot relate it intuitively in my mind to the concept that I’m working with (which is: to get the net change in the function…multiply the infinitesimal change in the independent variable(s) by the infinitesimal rate of change of the function with respect to that variable, and integrate). I would appreciate any help with this.

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