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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26 views

Implicit Differentiation Find dy/dx

Hi this question has got me stumped. It is to find dy/dx for the equation $$((x^2+y^2)^4+(x^4+y^4)^2)^{10}+x^5+y^5=100$$ I have done working to the following point ( if my working is correct ) : ...
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0answers
17 views

Solving applications of power series.

The region bounded by the curves $$ y = \frac{\sin(x^2)}{x} $$ $$ y = 0, x = \frac{1}{2}$$ is rotated around the y-axis. Find the volume of the solid of revolution with accuracy to within $10^{-2}.$ ...
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0answers
16 views

Double integral over an annulus

Question: Let $D$ be part of the annulus $1\le x^2+y^2 \le 4$ lying in the first quarter of the $oxy$ plane where $x \ge 0, y \ge 0$ and below the line $y=x$ Evaluate the integral $$\iint_D\ ...
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1answer
19 views

Evaluate the $\iint_S (x^2+y^2) \,dS$

I have to evaluate $\iint_S (x^2+y^2) \,dS$ where $S$ is the the portion of the sphere $x^2=y^2+z^2=4$ above $z=1$. Apologies for the formatting errors When I let $z=1, x^2+y^2=3$. So far I have: ...
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2answers
31 views

evaluating $\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy$

evaluating $\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy$ my book states $$\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy = \int_0^\infty \int_0^x y^2e^{-x^4} \ dy \ dx$$ Could someone ...
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0answers
11 views

Computing the normal vector when finding the flux

Use a parametrization to find the flux $\int \int_S F\cdot n$ $d\sigma$ across the surface in the given direction: $F=xy\overrightarrow i -z\overrightarrow k$ outward (normal away from the z-axis) ...
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0answers
11 views

Parametrising a side of a cuboid

Question: Suppose the surface S is bounded by 6 planes $$x=0,x=2,y=0,y=4,z=0,z=1$$ Parametrise two of the surfaces. My attempt: S0 I picked the "floor" face of the cuboid i.e. $x= 0, x=2, y=0,y=4, ...
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0answers
18 views

Stoke's Theorem?

Let $S$ be the portion of the plane $x+y+z=1$ that lies in the first octant, and let C be the boundary of S, traversed counterclockwise. Calculate $\int_{C} F.dr$ where ...
1
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1answer
17 views

Solving a surface integral using Gauss' Divergence Theorem

Question: Use Gauss's theorem to solve $$\iint_S F\cdot n~dS$$ given $$F(x,y,z)=(x,xy,z)$$ where S is the surface $$x^2+y^2= z^2, z \in [0,1]$$ My attempt: I have the solution and method for the ...
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1answer
15 views

Problem with multiple integrals and integration limits

I'm working in this integration. $\int_{1}^{2}\int_{1/Y}^{y}\sqrt{\frac{y}{x}}\left(e^{\sqrt{xy}}\right)dxdy$ I make this: $u=\sqrt{xy}$ ,$v=\sqrt{\frac{y}{x}}$, $x=\frac{u}{v}$, $y=uv$ For ...
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0answers
51 views

Find the limit: $\lim_{x\to 0,y\to0} \dfrac{(2x^2y)} {x^2+y^4} $

Find the limit: $\lim_{x\to 0,y\to0} \dfrac{(2x^2y)} {x^2+y^4} $ okay, I know the answer is 0. I can't understand or find why. please help.
2
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0answers
68 views

Functions $g(x)/h(x),h(x)/f(x)$ are constant [duplicate]

Suppose $f$, $g$, $h$ are functions from the set of positive real numbers into itself satisfying $f(x)g(y)=h(\sqrt{x^2+y^2})$ for all $x$, $y\in (0,\infty)$. Show that the functions $g(x)/h(x)$, ...
0
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1answer
11 views

Differentiation under integral sign proof question (special case)

The theorem in my textbook says: Suppose $f(s,x)$ and $f'_s(s,x)$ is continuous on $\alpha < s < \beta$, $a \leq x \leq b$. Then the function $$ \varphi(s) = \int_a^b f(s,x)\,dx $$ is ...
0
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1answer
25 views

Calculating a unit normal to the level surface at the point

I have a problem with this question: Calculate a unit normal $\hat n$ to the level surface $\phi=0$ ($\phi=x^2+y^2-z^2-1$) at the point r=$\hat j$ and sketch the level surface $\phi=0$. Your diagram ...
0
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1answer
26 views

Find a closed path $C$ such that $\oint_C {\bf F} \cdot d{\bf r} \neq 0$, where $F = (y^2,x,0)$

Consider the vector field $${\bf F}=(y^2,x,0).$$ Find a closed path $C$ such that $$\oint_C {\bf F} \cdot d{\bf r} \neq 0 .$$ My attempt: I decided to try with the unit circle however the ...
1
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1answer
20 views

Mass of lamina defined in y ≥ 0, with edges given by y = 0, y = (4-x^2)/3 and x = −y + 2y^2, and density is y.

I've been trying work this out, but I'm stuck on the the integral calculation. I've drawn a diagram, got all the points of intersection and relevant points, but I still can't get it. I had a go at ...
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2answers
22 views

$f(x, y) = \prod_{i = 1}^n (1 + xy_i)$, what is ${{{\partial f}\over{\partial x}}\over f}$, geometric series?

Let$$f(x, y) = \prod_{i = 1}^n (1 + xy_i).$$What is$${{{\partial f}\over{\partial x}}\over f}?$$What happens when we use the geometric series?
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2answers
30 views

Change integration limits, multivariable calculus.

Good night, i have a serious problem changing the integration limits, i read two books but i don't understand, i put an example... $\int_{0}^{1}\int_{0}^{1-x}\sqrt{x+y}\left(y-2x\right)^{2}dydx$ I ...
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0answers
27 views

Completing the Square of Quadratic Forms

I was working through a proof of a lemma that lets us determine whether a Hessian is positive definite for Mardens' Vector Calculus, page 175 Basically the lemma is if $B= \begin{bmatrix} a ...
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1answer
27 views

Finding the domain of the following integral in polar coordinates

Question: Convert the following integral into polar coordinates and solve $$\int_0^\frac{\sqrt{2}}{2}\int_x^\sqrt{1-x^2}xy \ dy\,dx$$ My attempt: I managed to get this: ...
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4answers
41 views

In Lagrange Multiplier, why level curves of $f$ and $g$ are tangent to each other?

In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise ...
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0answers
21 views

Problem with a triple integral for calculating a volume [duplicate]

I'm still struggling with this question: Calculate a volume using a triple integral By the order that user gave me it becomes easier but the question says to use that order in specific... Can anyone ...
2
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0answers
16 views

Deriving the curl of a vector field from the definition of torque.

I just learned about the definition of $\text{curl}\ F$ for some vector field $F(x, y)=M(x, y)\mathbf{e}_1+N(x, y)\mathbf{e}_2$ in $\Bbb{R}^2$ and was wondering how that could be derived from the ...
0
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1answer
11 views

Sign of a flux surface integral

Use a parametrization to find the flux $$\iint_S F \cdot n \, d\sigma$$ across the surface in a given direction: $$F=xy\overrightarrow i-z\overrightarrow k$$ outward (normal away from the z-axis) ...
1
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1answer
23 views

Sketching the surface $z=\frac{x^2y}{3}$

I am trying to sketch the part of $x^2+y^2=9$ which lies in the first octant between the surfaces $z=0$ and $z=\frac{x^2y}{3}$. I understand that $x^2+y^2=9$ is a cylinder with radius three, ...
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0answers
14 views

How to take partial derivative of a vector matrix vector multiplication?

I am trying to understand the mechanics of the below equations. I am especially confused about in 2.65 , how did the r.h.s which is a sum came from the gradient vector ? It would be great if someone ...
2
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1answer
42 views

Is this picture of the covariant derivative correct

I am reading O'Neil's Elementary Differential Geometry on my own. On page 81 he gave the following definition: Let $W$ be a vector field of $\mathbb{R}^3$, and let $v$ be a tangent vector field to ...
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0answers
12 views

interchanging a partial derivative and a summation in probability

Given this equation: $ \displaystyle\frac{\partial}{\partial \theta}\left . \left \{ -\dfrac{1}{N}\displaystyle\sum_{n=1}^N \ln p(\mathbf{x}_n|\theta) \right \}\right |_{\theta_{ML}} = 0$ ...
2
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1answer
14 views

Surface Normal from Cross Product

Given an equation (in this case $x^2-y^2+z=0$) how would I find the surface normal using a cross product at a certain $(1,2,3)$ point? I know how do it with $grad(f)$ but I presume that isn't what ...
4
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0answers
16 views

Can we detect smoothness of a norm by its behavior along paths?

We say a norm $\| \cdot \|$ on $\mathbb{R}^n$ is smooth if it is smooth as a function $\mathbb{R}^n\setminus \{0\} \to \mathbb{R}$. (i.e, after restricting the domain). We say a norm is smooth along ...
2
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2answers
31 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
6
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3answers
53 views

How to show that a continous function $f:\mathbb{R}^m \to \mathbb{R}$ has a maximum?

My task is this: Suppose $f:\mathbb{R}^m \to \mathbb{R}$ is a positive, continous function such that $\lim_{\mid \textbf{x}\mid \to \infty} f(\textbf{x}) = \textbf{0}$. Show that $f$ has a maximum. ...
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0answers
32 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
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1answer
23 views

How to set the limit for interated integral of $f(x,y)$ over diagonally partitioned region

I would like to compute $$I = \int_{\mathcal{R}} f(x,y) d\mathcal{R}$$ $$ f(x,y) = \begin{cases} x^2, \quad 0 < x < y < \pi \\ y^2 , \quad 0 < y < x < \pi \end{cases}$$ ...
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1answer
32 views

Problem with multiple integrals of $\cos(x+y)$

I have a problem with this integral $\int_{0}^{\pi}\int_{0}^{\pi}\mid \cos\left(x+y\right)\mid dxdy$ I work with this problem, but the result of the book does not match with my result Note: The book ...
1
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1answer
25 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions ...
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1answer
25 views

Finding the area between two curves using a set of transforms and their Jacobian

I have the following transforms: $\begin{align} x &= u^2 - v^2 \\ y &= 2uv \end{align}$ and am tasked with finding the area between the following curves: $\begin{align} x &= 4 - ...
2
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2answers
27 views

Langrange Multiplier, to find maximum volume of a cone

Question: A right-angled triangle is rotated about one of its sides that form the right angle to a cone. Given that the sum of the lengths of two sides of the triangle that form the right angle is ...
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0answers
8 views

Extremum of Laplacian of a function

Let f(x,y,z) be any arbitrary continuous function. Let's denote Laplacian of f by $\nabla^2 f$. 1) How do we denote the extremum of $\nabla^2 f$ mathematically ? 2) how do we solve for such ...
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2answers
31 views

Show that Set in $M:=\{x\in \Bbb R^3 : x_1^2\ge2(x_2^3+x_3^3) \}$ is closed

I have to show this regarding the Euclidean metric. I've already shown that it isn't bounded by showing that the $d(x,y)\:\forall x,y \in M$ isn't bounded. I know that in order to show the ...
0
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1answer
35 views

The region where the two variable function $xy/(x-y)$ is differentiable

I need to found the area where this function is differentiable $$ f(x,y) = \frac{xy}{x-y} $$ How do I need to proceed? For partial derivatives I got: $$ \frac{\partial f}{\partial x} = ...
0
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1answer
19 views

Volume 4-dimensional sphere

I'm studying Fubini's Theorem and Change of Variables Theore in class, and one of the exercises from last year exam was calculate the volumen of the 4D sphere. I searched on Internet how can I do that ...
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0answers
17 views

All possible paths to evaluate a multi variable limit

Most of the books that I have (H.K Dass) say that (or at least that's what I have understood) for the limit of a multivariable function (say f(x,y) ) to exist the limit along every possible path ...
1
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1answer
14 views

Surface integral of function over intersection between plane and unit sphere

I've been asked to compute the integral of $f(x, y, z)= 1 - x^2 - y^2 - z^2$ over the surface of the plane $x + y + z = t$ cut off by the sphere $x^2 + y^2 + z^2 = 1$ for $t \leq \sqrt3$ and prove it ...
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0answers
24 views

How to use the extreme value theorem on a vector valued function?

My task is this: Suppose that $A \subset\mathbb{R}^m$ is closed, bounded and that $\textbf{F}:A\to \mathbb{R}^k$ is continious. Show that $\exists ! \: K\in\mathbb{R}:\: ...
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1answer
13 views

Multivariate Normal cdf differentiation respect to dispersion

I am interesting in how to differentiate multivariate normal cdf respect to diagonal elements of covariance matrix (that is, I am interested only in variances). Problem similar to mine has been ...
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1answer
14 views

finding the expectation of the MLE for $\mathbf{\Sigma}$ in a multivariate Gaussian [on hold]

I am trying to find the expectation of the MLE for $\mathbf{\Sigma}$ for the multivariate gaussian. $E(\mathbf{\Sigma}_{ML}) = E\left (\dfrac{1}{N} \sum (\mathbf{x}_n - \mathbf{\mu})(\mathbf{x}_n - ...
0
votes
1answer
15 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
1
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0answers
10 views

Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
1
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0answers
14 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...