# Tag Info

1 vote

### How do I calculate the parametrization of a 3D surface, given its support function?

If $\vec u$ is a unit vector on the sphere and $h(\vec u)$ is the support function in that direction, then the expression $\vec p(u) = h(\vec u) \vec u - \nabla _S h(u)$ parametrizes your surface. ...
• 4,847
Accepted

### Strange usage of chain rule. Can anyone explain why this derivation was done this way?

(1.35) seems primarily motivated by geometry; if you follow their logic of $\delta x$ and look at the attached picture, it makes sense why they're doing it the way they did. If you want to explicitly ...

### Finding the area of an ellipse which is the intersection of $z=x^2+y^2$ and $z=1+2x$.

The ellipse can be written in parametric form as: $$x = 1 + \sqrt{2}\cos t$$ $$y = \sqrt{2} \sin t$$ $$z = 3 + 2\sqrt{2}\cos t$$ Where $x$ and $y$ form the $xy$-plane circle given by $(x-1)^2+y^2=2$, ...
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• 116k

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### Average distance from a point on a circle to the y-axis.

The average distance is what is made to the center of gravity of the disk from x-axis. $$x^2 +y^2=a^2$$ So we calculate $$\bar y= \frac{\int y^2 dx}{\int y~ dx}= \frac{Area First Moment }{Area}.$$
• 40.5k
1 vote
Accepted

### Average distance from a point on a circle to the y-axis.

On the circle $x^2+y^2=9$ we have identically $r\equiv 3$, so in the integral we should consider the length of the circle (which is $2\pi r=6\pi$) and integrate only on $\theta$. Thus, the integral is ...
• 5,366

### Finding the area of an ellipse which is the intersection of $z=x^2+y^2$ and $z=1+2x$.

Another solution comes from thinking of the projection of the ellipse onto the $xy$-plane. I realize that this is not the approach you were expected to take, but this might be useful to you in the ...
• 116k

• 1,585
1 vote
Accepted

### 'Integrating' a matrix times a gradient

Let $f=x^2-y^2$. And $$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.$$ Then $A \nabla f = \langle -2y,2x \rangle$ which has a non-zero curl, which implies it is not a gradient function. ...
• 8,186

### Area of a Quater-Circle with hyperbolic elements

Given the density function $D = xy$ and the quarter-disc in the first quadrant with radius $R$, we'll set up the integral to find the mass.$$\\$$Okay so mainly you can do that in 4 steps that you're ...

• 98.4k
Accepted

### Hessian matrix determinant greater than zero in a saddle point?

The second derivative test (and the Hessian determinant) only works for bivariate functions. For functions of three or more variables, one needs to use the eigenvalues. In this case, as we have both ...
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### Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$
Your notation suffers from a number of problems. (the least severe) $\varphi^{-1}$ can be replaced by the diffeomorphism $\psi$ to simplify notation (much more severe) the notation $\nabla$ does not ...