2

Is the process of variable substitution the same as function composition ? And if yes how do you write it in the case of multivariable functions ? For example a typical variable substitution is $$ x=r\cos(\theta), \ y=r\sin(\theta).$$ Given is $$f(x,y)=x^2 + y$$ Which is no turned into $$f(x(r,\theta),y(r,\theta))=g(r,\theta)=r^2\cos^2(\theta)+r\sin(\theta)$$ This looks to me like a composition of the function $$ f:\mathbb R^2 \rightarrow \mathbb R $$ and the function $$ p:\mathbb R^2 \rightarrow \mathbb R^2 $$ but i am unable to properly define $$p$$ which takes care of turning $(x,y)$ to $(r\cos(\theta), r\sin(\theta))$.

I seem to have a misconception some where but iam not able to make sense of it. Are changes of coordinate systems more than simple substitutions/function compositions or not ?

Hans Wurst
  • 211

1 Answers1

1

Yes, you are computing $f\circ p$ for $f$ given as you said and $p\colon \mathbb R^2\to\mathbb R^2$ given by $$ p(r,\theta) = (r\cos\theta,r\sin\theta). $$ Then $$ (f\circ p)(r,\theta) = f(p(r,\theta)) = f(r\cos\theta,r\sin\theta) = r^2\cos^2\theta + r\sin\theta. $$

Christoph
  • 24,912