# Tagged Questions

Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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### Reparametrization of a function

Given $$S(t)= \left( 1- \left( 1-{{\rm e}^{- \left( \rho\,t \right) ^{k}\theta}} \right) ^{\gamma} \right) ^{\frac{1}{\theta}}$$ where $\rho,k,\theta$ and $\gamma$ are ...
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### Find the surface area of that part of the cylinder

Find the surface area of that part of the cylinder given by $\mathbf{r}(u,v) = 3\cos u \mathbf{i} + 3\sin u\mathbf{j} + v\mathbf{k}$ over the region where $0\leq u \leq2\pi$ and $0\leq v \leq2$. The ...
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### Volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$

Find the volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$, using integration. It is clear that this is not centered at the origin. So, how do I find the limits for an integral? Any suggestion ...
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### Parameterize $\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}$

I need to parameterize the surface $$S=\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}.$$ My hint is that $S$ is a torus. I barely know where to begin. I have some idea on perhaps ...
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### Iterated Integral with variable substitution

I need to calculate the double integral of the function $f(x,y) = (x+y)^9(x-y)^9$: $\int_0^{1/2} \int_x^{1-x} (x+y)^9(x-y)^9 dydx$ I have a solution but I definitely arrived at it after a sloppy ...
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### Maximum and Minimums (Calculus 3) [on hold]

a)Find the maximum and minimums of the function on the indicated closed and bounded region R. f(x,y) = xy-2x R is the triangular region with vertices (0,0) , (0,4) , and (4,0).\ b) Find the ...
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### Directional derivative of a differentable function

Is it always true, for a differentiable function $F:\mathbb{R}^N\rightarrow\mathbb{R}^M$, that its directional derivative along a direction $v\in\mathbb{R}^N$ is equal to the product $J_f\cdot v$, ...
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### Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
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### How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
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### Equality of mixed partials proof

I'm trying to prove the equality of mixed partials. My book has a proof but it's only for functions $\Bbb R^2 \to \Bbb R$ (and then that can be extended to $\Bbb R^2\to \Bbb R^n$ by applying the ...
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### how can I prove this property is the necessary and sufficient condition [closed]

enter image description here Here is related formula! please click the link!
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### Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
Draw diagrams for cone ( with cone angle less than $360^{\circ}$) to show that the geodesics (generating ray and the warp around) have a projection of the curvature vector onto the tangent plane that ...