Find the absolute minimum and maximum values of $g(x, y) = (x^2 + y^2)e^{(−x^2−4y^2)}$ on the set $A = \{(x, y) \in \mathbb R^2 \mid x^2 + 4y^ 2 ≤ 4\}$.
here is the solution //see image
so my question is when you use parametric form how do you know what value to set for $x$ and $y$ given the boundary and why exactly did he choose $x=2\cos\phi$ and $y=\sin\phi$ instead of $x=\cos\phi$ and $y=\sin\phi$ any links to videos will be much appreciated
since the boundary is given by the form $x^2+4y^2≤4$ which can be written as $x^2+4y^2=4$ to parametrise an ellipse the following should be applied given the function for an ellipse $ (x-p)^2/(a)^2 + (y-q)^2/(b)^2 =1$ the parametrisation of the ellipse is as follows: $ a * cos(t)-p$ $ b * sin(t)-q$
so in this case since $x^2+4y^2=4$ since the right hand side isn't $=1$ we need to divide by whatever its value is to get it to equal 1 hence $ (x^2)/4+y^2=1$ this is in the form of an ellipse now and can be rewritten as $$(x-0)^2/(2)^2 +(y-0)^2/(1)^2 =1$$ which is equivalent to $$x^2/2^2 +y^2=1$$ and now applying the rule above we get the following $$x=2cos(t)$$ $$y=sin(t)$$ for more information regarding this topic please visit THIS YOUTUBE LINK