Functional relation between different functions of(x,y,z)

In summary, a functional relation between different functions of (x,y,z) is a mathematical relationship that shows how one function is dependent on another function or multiple functions. It can be expressed using equations, graphs, or tables and is important in understanding relationships between variables and solving real-life problems. A functional relation can be represented by a single function, known as a composite function, and can be determined by substituting the output of one function into the input of the other or by looking for patterns in their graphs or equations.
  • #1
souviktor
7
0

Homework Statement



I have two scalar functions u(x,y,z) and v(x,y,z) which are differentiable..Now it is required to prove that a necessary and sufficient condition for these two to be functionally related by equation F(u,v)=0 is [[tex]\nabla[/tex]u] [tex]\times[/tex] [[tex]\nabla[/tex]v]=0


The Attempt at a Solution


clearly the cross products of the gradients are zero that means they point in he same direction.But what about the tangent planes?and how to approach this problem?
 
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  • #2
0=F
0=grad(F)
0=(Fu)grad(u)+(Fv)grad(v)
so clearly we need
0=grad(u)xgrad(v)
 

1. What is the definition of a functional relation between different functions of (x,y,z)?

A functional relation between different functions of (x,y,z) is a mathematical relationship that shows how one function is dependent on another function or multiple functions. It represents how the output of one function changes based on the input of another function or multiple functions.

2. How can a functional relation between different functions of (x,y,z) be expressed mathematically?

A functional relation between different functions of (x,y,z) can be expressed using equations, graphs, or tables. For example, if we have the functions f(x) = x + 2 and g(y) = y^2, their functional relation can be expressed as f(g(y)) = (y^2) + 2.

3. What is the purpose of studying functional relations between different functions of (x,y,z)?

The study of functional relations between different functions of (x,y,z) is important in mathematics and other scientific fields as it helps us understand how different variables are related and how they affect each other. It also allows us to make predictions and solve problems in various real-life scenarios.

4. Can a functional relation between different functions of (x,y,z) be represented by a single function?

Yes, a functional relation between different functions of (x,y,z) can be represented by a single function, known as a composite function. This function combines the input and output of the individual functions to show their relationship.

5. How can we determine if two functions have a functional relation between them?

To determine if two functions have a functional relation between them, we can substitute the output of one function into the input of the other function. If the resulting value is equal to the output of the first function, then there is a functional relation between the two. Additionally, we can also look for patterns in the graphs or equations of the functions to determine their relationship.

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