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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1answer
41 views

Parametrization of a paraboloid part

Find the parametric equation of the surface $S$, where $S$ is the part of the paraboloid $z=x^2 + y^2 + 1$ bounded by the plane $z=2x+3$ My attempt The OXY projection of $S$ is $x^2 + y^2 + 1 = 2x + ...
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1answer
24 views

Surface integral over the plane $x+y+z=2$

Evaluate $$\iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy$$ where $S$ is the part of the plan $x+y+z=2$ in the first octant, with normal $n$ such that $n . (0,1,0) \geq 0$ My ...
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1answer
42 views

Transform rectangular region to region bounded by $y=1$ and $y=x^2$

I found this nice answer: But I am trying to find a way to map a rectangular region in the $uv$-plane onto the region bounded by $y=1$ and $y=x^2$ in the $xy$-plane. Any thoughts on how to do this?
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2answers
56 views

Does the set of Differentiable functions change if we change our norm?

This may be a naive question. I am reading the definition of differetiablity of a function $f:\mathbb{R^n}\rightarrow \mathbb{R^m}$ in the book Calculus Manifolds. I already know that all norms on ...
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0answers
99 views

Show that this function is continuous at $(0,0)$

In this case i'm struggling to show that the partial derivatives with respect to x are continuous. The answers always brush over how you determine it like it trivial so i think i'm missing something. ...
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0answers
180 views

Best book for learning multiple integrals, line integrals, Green's theorem, etc.

I've been searching for a book that teaches multiple integrals and such in a way that I can understand. I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
3
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2answers
66 views

Using Stokes's theorem to calculate a value of integral

Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where $\Gamma$ is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
2
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1answer
46 views

Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
0
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1answer
73 views

Surface integral over a sphere - parametrization

Evaluate the surface integral of the field $A(x,y,z)=(xy, yz, x^2)$ over the sphere $S$ givn by $x^2 + y^2 + z^2$ with the normal vector pointing to the exterior of the sphere I've tried doing this ...
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0answers
34 views

Volume of the set of two bodies

I have been given two bodys: $(x^2+y^2) < R^2(x^2-y^2)$ $x^2+y^2+z^2 < R^2$ Now I am supposed to calculate the volume of the set of the two bodies. I can see that the second body is a ...
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1answer
56 views

Triple intergration find the moment of inertia of a cylinder

What is the Moment of Inertia of a cylinder of Radius R, length l and mass M when rotated about an axis along the length of the cylinder? I've had a lot of examples of calculating the Moment of ...
3
votes
1answer
82 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
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0answers
45 views

Determine all points where 2 variable function is differentiable: when to use definition?

My function is: $$f(x,y) = \frac{x\sin^2 y}{x^2+y^2}\text{ when }(x,y) \ne (0,0)$$ and $0$ when $(x,y) = (0,0)$ When I use the definition (limit as $h$ approaches $0$), the limit for both partiales ...
1
vote
1answer
255 views

Finding out whether normal points outwards or inwards

Consider $F(x,y,z)=(x,x^2y,0)$ and $$\Omega=\{(x,y,z)\in\mathbb{R}^3\mid(x^2+y^2)^2<z<\sqrt{x^2+y^2}\}$$ I want to compute $\iint_{\partial\Omega}(F\cdot\nu)\text{ds}$ where $\nu$ is the normal ...
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1answer
66 views

Normal derivative

Let $f(x,y)=\frac{1}{2}\log(x^2+y^2)$ in $\mathbb{R}^2\setminus (0,0)$. Is the limit of normal derivative of $f$ with respect to the unit circle ($\mathbb{S}=\{(x,y): \: x^2+y^2=1\}$) as $(x,y)$ ...
2
votes
1answer
54 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
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0answers
46 views

Gauss' universal theorem

I recently came across a set of lecture notes on multivariable calculus. In a section about integral theorems, the standard Gauss' (divergence) theorem was brought up: $\int_{V}\nabla \cdot ...
2
votes
1answer
113 views

This double integral

$$ \int_0^1\int_0^1x^3y^2\sqrt{1+x^2+y^2}\hspace{1mm}dxdy$$ We have to compute this up to 4 decimal places
1
vote
1answer
329 views

Multiple Integrals: Moment of inertia of a cylinder

Question: Calculate the moment of inertia of the cylinder defined below when the cylinder is rotated around the $x$-axis. The cylinder’s axis lies along the z-axis and is defined by $x^2+y^2=1$, $z ≥ 0$ ...
0
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1answer
51 views

How to find the boundaries of integration for e.g. triple integrations?

I'm having a lot of trouble finding from where to where I have to integrate when splitting up a triple integral into 3 integrals. I've already posted a question regarding this but while that helps ...
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2answers
99 views

Calculating $\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$.

I need to calculate the following in cylindrical coordinates: $$\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$$ $K$ is bounded by the plane $z=3$ and by the cone $x^2+y^2=z^2$. I know that: ...
2
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0answers
26 views

help finding this surface integral

Given that $$S : z = xe^y, \hspace{1mm} 0 \leq x\leq 1, \hspace{1mm} 0 \leq y\leq 1 $$ Find $$\int\int_S \left(x^2+y^2+z^2\right)dS$$ upto four decimal places
2
votes
1answer
32 views

A question on multivariate integral

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a given function. Suppose $\boldsymbol{f}:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ is the vector version of $f$, e.g., ...
1
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1answer
48 views

How do I prove continuity of partial derivative of $f(x, y) = \sqrt{x^4+y^4}$ at $(0,0)$?

Consider I have $$f(x, y) = \sqrt{x^4+y^4}$$ And I want to check if the function has partial derivatives continuous in point $$(x_0, y_0) = (0, 0)$$ I know theorem, that existence of continuous ...
2
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1answer
47 views

How to show that this equation involving partial derivatives is true? (Change of variables)

Let $z$ be a function of $u$ and $v$ where $u=x+y$ and $v=3x-3y$. I have previously shown, by the Chain Rule, that $$\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}= \left(\frac{\partial ...
2
votes
1answer
47 views

Proving a theorem on limits

I need to prove that: If $$\lim_{(x,y) \to (a,b)}f(x,y)= 0 \text{ and } g(x,y)\leq k,$$ then: $$\lim_{(x,y) \to (a,b)}f(x,y) g(x,y) =0.$$ My approach is like follow: ...
0
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1answer
34 views

symetry of partial derivative

If $f : \mathbb{R}^{n} \to \mathbb{R}$ has continuous second partial derivatives, then $\forall i, j $ $$\frac{\partial^2 f(x_1, \dots, x_n)}{\partial x_i\, \partial x_j} = \frac{\partial^2 f(x_1, ...
0
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1answer
23 views

Problems regarding multivariable calculus

Let $f:\Omega\to \mathbb R$ be differentiable at $x_0\in \Omega$ ($\Omega$ is a nonempty open subset of $\mathbb R^n$), let $f(x_0)=0$ and let $g:\Omega\to \mathbb R$ be continuous at $x_0$. We want ...
4
votes
1answer
180 views

Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
0
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1answer
44 views

Multiple Integration: where's the mistake in my process?

Evaluate: $$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where D is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ...
0
votes
1answer
49 views

Gradient descent with adaptive learning ratio.

I have a neural network, trained with SGD (stochastic gradient descent) with learning ratio $\alpha$. Each iteration I try to recalculate the weights with a rule: $$\Delta \vec{w} = -\alpha ...
2
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0answers
102 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
0
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1answer
38 views

Define multiple-variable function to be continuous

Define the function $f(x,y)= {{x^2 + y (x^2 + y)} \over {x^2 + y^2}}$ at $[0,0]$ so that the function would be continuous. I need help with this calculus problem. I mean, I guess it involves some ...
0
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1answer
33 views

Proving the pseudosphere is regular and orientable.

The textbook I'm using define the tractrix by $ T=\{(\sin t, \cos t+\log (\tan (t/2))):0<t\leq\pi\}$ and define the pseudospher being the tractrix roting around the $z$-axis, I have to prove that ...
3
votes
2answers
145 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
0
votes
1answer
119 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
3
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0answers
36 views

Evaluating a triple integral explained step by step

Evaluate: $$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where $D$ is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ...
0
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1answer
429 views

Surface integral - The area of a plane inside a cylinder

I'm having trouble with this question: Find the Surface Area of the part of the plane $x+2y+z=4$ that is inside the cylinder $x^2+y^2=4$. I tried writing the surface like this: $$(r\cos(\theta), ...
2
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1answer
77 views

Equation for the tangent plane and the normal line of $f(x, y, z) = x^2 + y^2 + z$

I have question: Find the equation for the tangent plane and the normal line of the surface $f(x,y,z)=x^2+y^2+z$ at point $(1,1,1)$ For the tangent plane I got, $z=2x+2y+z-2$ is this correct, ...
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1answer
64 views

Evaluating $\int_{\mathbb R^n}e^{-\langle Ax,x \rangle}dx$ where $A$ is symmetric and positive definite [duplicate]

I was asked the following question, and while I think I made little progress, I'd like a push in the right direction. Let $A$ be $n$x$n$ positive definite symmetric matrix with real entries. then for ...
4
votes
0answers
81 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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1answer
444 views

Sum of concave and strictly concave functions

How can I prove, for the general case, that the sum of a concave and a strictly concave function, yields a strictly concave function?
3
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1answer
43 views

Can the divergence theorem be restricted to flat surfaces?

I am trying to prove this: $S$ is a bounded surface in $\mathbb{R}^2$ and $a$ a given scalar field $u,v$ are such that $\nabla^2 u=0$ on $S$ and $u=v=a$ on $\partial S$. Then: $$\int_S ...
2
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1answer
47 views

Smoothness of $f(x)/(1+|f(x)|)$ where $f \in C^1(E)$ for $E$ an open subset of $\mathbb{R}^n$

(a) Show that if $E$ is an open subset of $\mathbb{R}$ and $f \in C^1(E)$ then the function $$F(x) = \frac{f(x)}{1+|f(x)|}$$ satisfies $F \in C^1(E)$. (b) Extend the results of part (a) to $f \in ...
0
votes
1answer
46 views

Integrating in spherical polar coordinates

Given a function $f(r,\theta,\phi)$ expressed in polar coordinates, would its integral over a sphere of radius $R$ centered at the origin be: $$\int_0^{2\pi}\int_0^{\pi}\int_0^R ...
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vote
1answer
20 views

Solution check: integral calculation using transformations

Could somebody check my solution to the following problem? Calculate $\iint_Ey\cos(y^2-4-x)dxdy$ if $E \in ]-\infty,0]\times [0,+\infty[$ enclosed by $x=-2, y=0, y^2-4=x$. Using $u=y^2-3,\quad ...
0
votes
1answer
73 views

Evaluate a triple integral over an ellipsoid.

Evaluate $$\iiint_{T} \sqrt{(1-9z^2)(1-4y^2-9z^2)}\, dx\,dy\,dz,$$ where $T$ is the domain $$x^2+4y^2+9z^2\le1.$$ I'm not sure of my result. Can you write a solution step-by-step please? ...
2
votes
2answers
228 views

Find volume of sphere $ x^2+y^2+z^2=9$ bounded by planes $z=0$ and $z=2$ using double integral

Find volume of sphere $x^2+y^2+z^2=9 $ bounded by planes $z=0$ and $z=2$ using double integral I tried to take the total volume of the bigger hemisphere but i get zero, i managed to take the volume ...
0
votes
1answer
63 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
0
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1answer
64 views

Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...