Implicit functions in R2 vs. functions of two variables in R3

In summary: Bear,In summary, the conversation discusses the concept of implicit differentiation and how it relates to functions of two variables. The speaker uses the example of a circle of radius one and a 3D equation to explain how partial derivatives can be used to find the derivative of one variable in terms of another, while holding the third variable constant. The conversation also touches on the significance of setting a variable to a constant in terms of finding partial derivatives.
  • #1
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Hello,


I often encounter some confusion.


For example, we have the equation for a circle of radius one given by:

x2+y2=1


Now we can't express this as a function in R2 because it's a relation. however, we can still find derivatives like dy/dx or dx/dy using implicit differentiation.


Something that gets me though is, how does this implicit function relate to a function of two varaibles? For example, we could create the function

z=x2+y2


Now for example if we let z=1, won't this be the circle of radius one, lying in the plane z=1?


How do the derivatives of these two things relate to each other? For example, in multivariable equations, we wouldn't just say "what's the derivative of z" because that doesn't make sense, we'd think, "well, in terms of what variable?". If we wanted dz/dx we'd take the partial derivative of z in terms of x. But for example, if we go back to the 2D relation x2+y2=1, if we want to find the derivative of y in terms of x, we use implicit differentiation and go:

d/dx(x2+y2)=d/dx(1)

2x+2y(dy/dx)=0 --> (dy/dx)=(-x/y)


But if we now go back to our 3D equation z=x2+y2, say we want the derivative of y in terms of x when z = 1. Would this end up being the same thing?
 
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  • #2
Partial derivatives hold either x or y to be constant so you can see how the height of the surface Z varies with only a single variable.

In your example of a circle, it is simply Z assigned to be a constant 1, in which case the partial derivatives of Z has no significance, since Z is no longer a function of the other two.

However, if you take [itex] z^{2} = x^{2} + y^{2} [/itex] and solve it for y, you get [itex] y^{2} = z^{2} - x^{2} [/itex]

In 3 dimensions, this is still the exact same surface, but it has y as the subject. Now you can find partial derivatives of y with respect to x holding z constant!
So you can find

[tex] \frac{∂y}{∂x} = \frac{-2x}{\sqrt{z^{2}-x^{2}}} [/tex]

This basically tells you how y changes with respect to x, holding z to be a constant. In your case, z=1, so we set z=1.

With some algebra, we get

[tex] \frac{∂y}{∂x} = \frac{-x}{y} [/tex] when z=1

BiP
 

1. What is the difference between an implicit function in R2 and a function of two variables in R3?

An implicit function in R2 is a function that cannot be explicitly written as an equation with one variable isolated on one side. Instead, it is defined by a relationship between two variables. On the other hand, a function of two variables in R3 is a function that takes two inputs and produces one output, and can be explicitly written as an equation with both variables present.

2. How are implicit functions and functions of two variables used in real-world applications?

Implicit functions are often used to model relationships between variables in situations where it is not feasible to explicitly solve for one variable in terms of the other. Functions of two variables, on the other hand, are used to represent physical quantities or phenomena that depend on two independent variables. They are commonly used in fields such as physics, engineering, and economics.

3. Can an implicit function be graphed in two dimensions?

No, an implicit function cannot be graphed in two dimensions because it does not have a single, explicit equation. Instead, it is represented as a curve or surface in three dimensions, with the variables x and y as inputs and the relationship between them defining the shape of the curve or surface.

4. How are implicit functions and functions of two variables related?

Implicit functions and functions of two variables are related in that they both involve two variables and a relationship between them. However, implicit functions have a more general form and can be used to describe a wider range of relationships, while functions of two variables have a more specific form and are used to represent a specific mathematical function.

5. What are some common examples of implicit functions and functions of two variables?

A common example of an implicit function is the unit circle, which is defined by the equation x^2 + y^2 = 1. A common example of a function of two variables is the height of a projectile as a function of time and initial velocity, which can be represented by the equation h(t) = -0.5gt^2 + v0t + h0, where g is the acceleration due to gravity, v0 is the initial velocity, and h0 is the initial height.

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