Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

So I am taking mathematical economics and in the HW my professor asked to draw a couple of level curves for $f(x,y)=xy$.

Attempt: So I did it the following way. To find the slope I took the partial derivative with respect to $x$ and then with respect to $y$.:

$$\frac{\partial f}{\partial x}=y\\\frac{\partial f}{\partial y}=x$$

Now to find $\frac{\partial y}{\partial x}$ I just divide the upper equality $(\frac{\partial f}{\partial x})$ by the lower equality ($\frac{\partial f}{\partial y}$):

$$\frac{\partial f}{\partial x} \frac{\partial y}{\partial f}=\frac{\partial y}{\partial x}=\frac{y}{x}$$

So why is it wrong and why should I use the total differential to find the correct slope? Any hints please.

share|improve this question
1  
I don't know why you are focusing somewhat indirectly on finding the slopes of these curves, when you can compute their equations explicitly. Level curves of $f(x,y)$ are curves implicitly defined by equations of the form $f(x,y) = c$ for $c$ constant. So you could draw, for example, graphs of the curves $xy = 1$, $xy=-1$, $xy = 5$, etc. (When $c \neq 0$ the level curve for the value $c$ is a hyperbola with equation $y = c/x$. When $c = 0$ the level curve is the union of the two axes.) –  leslie townes May 26 '12 at 23:14
    
yeah I know that. But, how can I find the slope of the level curves? Namely, $\frac{\partial y}{\partial x}$. It is a part of the assignment. –  Koba May 26 '12 at 23:17

1 Answer 1

up vote 2 down vote accepted

Level curves have $f(x,y)=C$ for some constant $C$. So you can take: $$C=xy$$ $$y=\frac Cx$$ and then differentiate to get $$\frac{dy}{dx}=-\frac{C}{x^2}$$

share|improve this answer
    
Alright. What if i want to skecth level sets or in other words the contour plots on the xy-plane? How do I find the slope of the contour plots? –  Koba May 27 '12 at 4:58
1  
This is the slope of the contour plot. You have $z=f(x,y)$. $\frac {\partial z}{\partial x}$ is in the xz plane, $\frac {\partial z}{\partial y}$ is in the yz plane. A level set is in the xy plane, so to get the slope along a level curve you want $\frac {dy}{dx}$. We're dealing with partial derivatives here: $\frac {\partial z}{\partial x} \neq \frac {dz}{dx}$, so the chain rule doesn't work the way you used it. If you want to sketch contour plots i'd just sketch $y=C/x$ for evenly-spaced values of C directly. I don't see why you'd want/need the derivative. –  Robert Mastragostino May 27 '12 at 16:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.