# Questions tagged [multivalued-functions]

In Mathematics, set theory, a Multivalued function is defined as a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.

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### Is this function from $\mathbb{R}^3$ to $\mathbb{R}^4$ locally invertible at $(x,y,z)$?

Is this function from $\mathbb{R}^3$ to $\mathbb{R}^4$ invertible at $(x,y,z)$? $k(x,y,z) = (x+y+z, e^x \cos z, e^x \sin z, \cos z)$ at $(x,y,z)$ I have only studied the Inverse Function Theorem ...
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### An exercise in high-dimensional chain rule and Jacobians

In my vector calculus class, we are studying multivariate vector-valued functions and I have come across this exercise Let $f: \mathbb{R}^2 \to \mathbb{R}^3$, $g: \mathbb{R}^3 \to \mathbb{R}^2$, and ...
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### Product and set division in multivalued function Complex Analysis

i know that $a^{\frac{1}{n}}=\lbrace z\in \mathbb{C}: z^n=a \rbrace$, now consider the following: $[(-1)^{\frac{1}{4}}]^2=[\lbrace 1, i, -1, -i\rbrace]^2=\lbrace 1,-1\rbrace$ the question is Why? ...
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### If the line integral of a differential = 0 does this imply it is a total differential?

Assume a differential of the form $df(x,y) = X(x,y) \,dx + Y(x,y) \,dy$. If $\oint\ df=0$, it's easy to see that $\frac{\partial X}{\partial y} = \frac{\partial Y}{\partial x}$, which can be seen ...
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### Find local extremum and saddle points of the function $f(x, y) = x^2y^3(6 − x − y)$

$\nabla \left(f\right) = \begin{bmatrix}y^3x\left(12-3x-2y\right)\\ x^2y^2\left(18-3x-4y\right)\end{bmatrix}$ and in $(0,a), (a,0), (2,3)$ points $\nabla \left(f\right) = 0$ for all real a. Hessian ...
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