Questions tagged [implicit-function]
This tag is for questions relating to "implicit function", a function or relation in which the dependent variable is not isolated on one side of the equation.
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Signed Distance Functions into explicit representations?
I am creating a product which would benefit from conversion of 3d implicit surfaces (also called Signed Distance Functions or F-reps) to explicit boundary representations.
I am creating implicit ...
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Finding critical points of a Morse function on an implicit manifold
In the special case that a manifold is given by some implicit equation
$$ F(x_0, x_1, \dots, x_N) = 0$$
Then a natural Morse function is to simply take $f(p) = X_i$, where $p = (X_0, X_1, \cdots X_i, \...
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Implicit differentiation choice
I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows,
Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
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Second order partial implicit derivatives
I have some function $F(x, y, z) = 0$, and wish to find the second order cross derivative
$\frac{d^2z}{dxdy}$.
I've easily been able to get the second order derivatives $\frac{d^2z}{dx^2}$ and $\frac{...
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Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,005
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Find the area enclosed by the curve $(x^2+y^2-1)^3-x^2y^3=0$
The curve
$$(x^2+y^2-1)^3-x^2y^3=0$$
forms a heart shaped curve.
I want to find the area enclosed by it.
This curve is a sixth degree algebraic curve, so y cannot be found and x cannot be found. The ...
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Condition on derivatives
I am working with a "well-behaved" optimization problem of the form:
\begin{equation*}
\max_{x} f( g_{1}( x) ,g_{2}( x) ,g_{3}( x) ,\mathbf{y})
\end{equation*}
where $\displaystyle f:\mathbb{...
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What is the scalar field the integral of which gradient norm is equal to the surface area of the surface that the scalar field represent?
Assume I have a scalar function $\phi(p): p \in \Omega \subset \mathcal{R}^3 \to \mathcal{R}$. I would like to use it to represent a 2D surface. For example, if $\phi(\cdot)$ is a signed distance ...
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Given $ye^{-{xe^{y-2x}}} = 2xe^{-x},$ where $x>\frac{1}{\sqrt{2}}, y<\sqrt{2}$. Show that $y=y(x)$ decreases.
Suppose $y$ is defined by the following implicit equation: $ye^{-{xe^{y-2x}}} = 2xe^{-x},$ where $x,y\geq 0.$
I want to show that $y$ decreases as $x$ increases, when $x>\frac{1}{\sqrt{2}}$ and $y&...
user1288879
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Finding the distance of the intersection point of two conics to $(-3,2)$
Consider the intersection of the curves $x^2+y^2+6x-24y+72=0, x^2-y^2+6x+16y-46=0$. Determine the sum of the distances of their intersection points and $(-3,2)$.
My first thought looking at this ...
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Solving explicitly $y(x)$ in an exact ODE
Given the following differential equation:
$$y^2+(2xy+1)\frac{dy}{dx}=0$$
I have found that the solution is given implictly by:
$$ (\ast) \quad xy^2(x)+y(x)=k \: ; k \in \mathbb{R}$$
If I was ...
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Implicit equation of all points that a circle that traces along a 2d parametric curve.
I want to find an implicit equation that contains points that fall within a circle that has an origin that follows a 2d parametric curve, which would look like you painted a circle along that curve. I ...
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Implicit equation of revolution $(x^2+y^2)^2=x$
I have a curve given as $(x^2+y^2)^2=x$ in the plane $z=0$. We then rotate this curve around the x axis and then must find a parametrization for the surface as well as find the tangent plane in the ...
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Predictor Corrector Scheme for Implicit Runge Kutta
I want to solve an ODE system :
$$
\frac{dy}{dt} = f(y, t)
$$
Since my application requires method to be symplectic, I am using an implicit runge kutta method.
$$
y_{n+1} = y_n + h\sum_{i=1}^s{b_iK_i}
...
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Implicit equation of the limacon
I am studying from the textbook Elementary Differential Geometry by AN Pressley (2nd edition).
At the end of the first chapter the Author discusses the relationship between level curve and ...
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