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Physics_wiz
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I have a simple question, let's say I have a function f = f(h(a,b), c, d). Can I express this as f = g(a, b, u(c,d))? Are the two expressions equivalent or is one different/more general than the other?
jostpuur said:The following claim should be true, perhaps it answers something:
For arbitrary function [tex]f:\mathbb{R}^4\to\mathbb{R}[/tex], there exists functions [tex]g:\mathbb{R}^3\to\mathbb{R}[/tex] and [tex]u:\mathbb{R}^2\to\mathbb{R}[/tex], so that
[tex]
f(x_1,x_2,x_3,x_4) = g(x_1,x_2,u(x_3,x_4)),\quad\forall\; x_1,\ldots,x_4\in\mathbb{R}.
[/tex]
The reason for this is that [tex]\mathbb{R}[/tex] and [tex]\mathbb{R}^2[/tex] have the same cardinality, so that there exists a bijection [tex]u:\mathbb{R}^2\to\mathbb{R}[/tex].
Physics_wiz said:I don't think I agree with that.
Let [tex] f(x_1,x_2,x_3,x_4)=x_1x_3+x_4[/tex]. Now, what function [tex]g(x_1,x_2,u(x_3,x_4))[/tex] is equal to [tex]f(x_1,x_2,x_3,x_4)[/tex]
Physics_wiz said:I have a simple question, let's say I have a function f = f(h(a,b), c, d). Can I express this as f = g(a, b, u(c,d))? Are the two expressions equivalent or is one different/more general than the other?
HallsofIvy said:In general, f(h(a,b), c, d) can be written as k(a,b,c, d) but cannot be written as g(a,b,u(c,d)) since the last assumes that c and d appear throughout f only in a specific form: u(c,d).
Physics_wiz said:Ok, here's where the original question came from...maybe this helps.
Say I have a function [tex]F(a,b,c) = G(d,e)[/tex]. Assume the Implicit Function Theorem conditions are satisfied. So, I can solve for c as follows: [tex]c = H(G(d,e),a,b)[/tex]. Now, in this case, can I write c as [tex]c = N(d,e,M(a,b))[/tex]? Why or why not?
A multi-variable function is a mathematical concept that involves more than one independent variable. In other words, the output of the function is dependent on multiple input variables, rather than just one.
A multi-variable function is typically expressed using mathematical notation, with the input variables listed inside parentheses and the output variable after an equal sign. For example, the function f(x,y) = x + y would take two input variables (x and y) and output their sum.
Expressing a multi-variable function allows us to understand and analyze the relationship between the input variables and the output variable. It also allows us to make predictions and solve problems involving multiple variables.
No, not all multi-variable functions are linear. A linear function is one in which the output variable changes at a constant rate in relation to the input variables. However, multi-variable functions can be non-linear, meaning the output variable changes at a non-constant rate in relation to the input variables.
Yes, a multi-variable function can have any number of input variables. The number of input variables is determined by the specific problem or situation being analyzed. For example, a function that calculates the volume of a box would have three input variables (length, width, and height).