So I think I understand how to use the Implicit Function theorem to find partial derivatives given one function but I am confused as to how to do this for 2 functions. I’m trying to find $\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}$ around the point $(1,-1,-1,2)$ given the equations $$x^2+2y^2+u^2+v=6$$$$2x^3+4y^2+u+v^2=9$$ and I thus calculated that […]

While an explicit function $y(x)$’s arc length $s$ is easily obtained as $$s = \int \sqrt{1+|y'(x)|^2}\,dx,$$ is there any formula for implicit functions given by $f(x,y) = 0$? One can use the implicit differentiation $y'(x) = -\frac{\partial_y f}{\partial_x f}$ to obtain $$s = \int\sqrt{1 + |\partial_y f / \partial_x f|^2}\,dx,$$ but that still requires (locally) […]

In C.H. Edward’s Advanced Calculus of Several Variables in the Chapter III in Section 3 on Inverse and Implicit Mapping Theorems question #5 is given as follows: 3.5 Show that the equations $$ \sin(x+z)+\log yz^2 = 0, \qquad e^{x+z}+yz=0 $$ implicitly define $z$ near $-1$ as a function of $(x,y)$ near $(1,1)$. My trouble here […]

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$ Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of $z$, and find $\frac{dx}{dz}$ and $\frac{dy}{dz}$ What I did: The differential of $F$ is: $\begin{pmatrix} 1 & z &y-3z^2\\3x^2-z & 3y^2 &-x\end{pmatrix}$, insert $x=1,y=-1,z=0$ to get: $\begin{pmatrix} […]

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for examples of this, but all I have found is the wikipedia article for the topic and the examples there were too rushed(or requiring too […]

Trying to understand parametric and implicit line equations but I’m at a complete halt now. I have a line that goes through P(0,1) and Q(3,2), I need to find the implicit equation N((x,y)- Z) = 0 such that Z is a point on the line and N is a vector perpendicular to the line. Please […]

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y \to 0} \frac{\sin y}{y} = 1$ I want to say that in the vicinity of $(0,0)$ $y=\sin y$ and then we get […]

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