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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27 views

Finding bounds of integration

Let $S$ be the region in the first quadrant of the $xy$-plane bounded by the $x$-axis and the parabolas $$x=1-\dfrac{1}{4}y^2,$$ $$x=\dfrac{1}{4}y^2-1$$ and $$x=4-\dfrac{1}{16}y^2.$$ Use the ...
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2answers
40 views

changing order of integration

I was tasked with changing the order then drawing and calculating the integral $\int _0^2 dy\:\int _{y}^{y+2}\:\frac{x}{y+2}dx$...which got very complex. I understood that the D area had to be split ...
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4answers
107 views

The integral of Gaussian function of three variables [closed]

How do I solve this $$\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty} e^{-x^2-y^2-z^2}\ dx \ dy \ dz$$
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1answer
105 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
3
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3answers
75 views

How to prove $\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$?

To prove that $$\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$$ I start with $$\left| {\frac{{{x^3}{y^2}}}{{{x^4} + 3{y^4}}}} \right| \leqslant \left| ...
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1answer
59 views

Partial derivatives of $\ln(x^2+y^2)$

I am new to partial derivatives and they seem pretty easy, but I am having trouble with this one: $$\frac{\partial}{\partial x} \ln(x^2+y^2)$$ now if this was just $\frac{d}{dx}\ln(x^2)$ we would get ...
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2answers
44 views

Limit in multivariable-calculus

Let $\ell$ be a straight line through origo. Determine the limit to the restriction of $$f(x,y)=xye^{-x^2y^2}$$ to $\ell$ when $x^2+y^2 \to \infty$. Also, investigate the limit $$\lim_{x^2+y^2 \to ...
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1answer
22 views

Prove the Jacobian identity

How to prove that these Jacobians are equal? $$\dfrac{\partial (x,y)}{\partial(\alpha, \beta)} \cdot \dfrac{\partial(\alpha, \beta)}{\partial(z,w)} = \dfrac{\partial (x,y)}{\partial(z,w)}$$ I don't ...
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4answers
59 views

How can I prove that any ball in $\mathbb{R}^n$ is connected?

As the title follows, how can I prove that any ball in $\mathbb{R}^n$ is connected? or can you give me a hint? I have some ideas but I'm not sure about them. I thank any help you can give me! ...
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1answer
23 views

Is the set of all $(x,y,z)$ such that $z^2-x^2-y^2-1 = 0$ open or closed?

As the title says, is the set of all $(x,y,z)$ such that $z^2-x^2-y^2-1 = 0$ open or closed? Moreover, how can I prove it? I understand the definition of open and closed sets, but I don't get this ...
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1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
2
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2answers
21 views

Difference between path and vector field

What is the difference between a path and a vector field? From what I understand the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields (constant vector fields to ...
4
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2answers
113 views

solution of $y^{\prime \prime} + y^n = 1$ [closed]

I am not able to figure out the solution for the differential solution $$y^{\prime \prime} + y^n = 1$$ I want to specifically find an answer for $$y^{\prime \prime} + y^2= 1$$and $$y^{\prime \prime} + ...
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0answers
16 views

Justifying the “Dual feasibility”, one of the Karush-Kuhn-Tucker conditions

I am having difficulty of interpreting the KKT conditions in a general setting where we have $M$ equality and $N$ inequality constraints defined as: Minimize $f(x)$ subject to $g_i(x) \leq 0 , h_j(x) ...
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3answers
55 views

Find partial derivatives of $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=x^3+y^3+z^3$

I've been trying to solve this question using the Implicit functions theorem from Schaum's outline series (Theory and Problems of Differential and Integral Calculus, by Frank Ayres) with no luck: ...
0
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1answer
64 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
0
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1answer
20 views

Are Gradients of Radial Functions parallel?

For two radial functions $f(R)$ and $g(R)$ defined on an open set $U \subset \mathbb{R}^2$, is $\nabla f$ always parallel to $\nabla g$? Since both functions are constant around each circle $C_r(0)$ ...
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1answer
37 views

Region bounded by $x=y^2$ and $x=y^3$

What is meant by "the region bounded by $x=y^2$ and $x=y^3$"? The graphs of these two curves split the plane into 4 sections, but none of these are really bounded, they all kind of continue forever in ...
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0answers
25 views

Partial derivative of $F{\equiv}0$

If you have a function $F{\equiv}0$ then is the partial derivative of $F$ with respect to any of its variables $0$? Specifically, when we have Charpit's equations for a PDE $F(x,y,u,p,q) = 0$, where ...
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1answer
35 views

How can I go about solving this group of equations in as simple a way as possible?

They arise from partial derivatives of the Lagrange multiplier function. Here below is the original problem: Goal function: $$f(x,y,z)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} $$ with ...
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1answer
56 views

Fundamental Theorem of Calculus With Function Containing Limit Variable

I'm trying to solve the following question: Evaluate $$\frac{\mathrm{d} }{\mathrm{d} s} \int^s_0 e^{st^2} dt $$ My thinking was that by the fundamental theorem of calculus, we have $ F(s) = ...
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0answers
25 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
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2answers
55 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
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0answers
28 views

A problem of vector integration: Show that $\iint_S f grad f \times dS =0$

For any scalar field $f$, show that $\iint_S f\, \nabla f \times dS =0$. I don't have an idea to solve. Please help me.
2
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1answer
37 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
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1answer
31 views

Proving that the norm of $f'(y)$ is attained at $\pm\frac{\nabla f(y)}{\|\nabla f(y)\|}$.

Consider a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$ and a point $y\in \mathbb{R}^n$ such that $\nabla f(y)\neq 0$. Prove that there exists an unit vector $x_0\in\mathbb{R}^n$ such that ...
4
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3answers
112 views

Proof of this definite integral?

Saw this sometime in my calculus book, from the Putnam Math Challenges listed: $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ ...
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3answers
49 views

How to prove that the following function is continuous on $\mathbb{R^2}$

$$f(x,y)=\begin{cases} \dfrac{x^2y}{x^2+y^2}\Leftrightarrow x^2+y^2\not=0\\0\Leftrightarrow x=y=0\end{cases}$$
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1answer
14 views

Transversals Related to Circles and Spheres

I was wondering if anyone could provide insight into whether the intersection of 3 circles (no interior) in R^2 intersecting at a single point would be transversal. My struggle in understanding some ...
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3answers
32 views

Limit of several variables

What would be the limit of the given multi-variable functions? $$1. \lim_{(x,y) \to (0,0)} x \cdot sin \left(\frac{1}{y}\right)$$ $$2. \lim_{(x,y) \to (0,1)} \frac{e^{x}-y}{xy}$$ I tried to solve ...
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1answer
42 views

Show that this is indeed a differentiable manifold with boundary.

I want to show that the cylinder: $$C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$$ is indeed a a differentiable manifold with boundary, this means the following: A subset $M ...
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2answers
54 views

Multivariable limit which should be simple !

How to calculate the following limit WITHOUT using spherical coordinates? $$ \lim _{(x,y,z)\to (0,0,0) } \frac{x^3+y^3+z^3}{x^2+y^2+z^2} $$ ? Thanks in advance
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3answers
55 views

Finding the area of an ellipse

Using Green's Theorem, find the area of the ellipse $\frac{x^2}9+\frac{y^2}{16}=1$. My work so far: Green's Theorem states that $\iint_R \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial ...
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1answer
36 views

Computing the Limit of Multivariable function

How do we compute the limit of following function? $$\lim_{(x,y,z) \to (0,0,0)} \frac{sin(x^2+y^2+z^2)}{x^2+y^2+z^2}$$ If someone can give me some hint then that would be great. Thanks.
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2answers
24 views

Evaluating limit of multi variables

The question: $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(x+y)}{x+y}.$$ I tried this using $y=x$ path and $y=0$ path and both approach to the same value, $1$. My problem is to show that this value really ...
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0answers
22 views

Finding points on a surface which contain tangent planes parallel to the axis-planes

Suppose that $f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z}$ and let $S$ be the surface given by the equation $ f(x,y,z) = 1 $ Are there any points on $S$ where the tangent plane to $S$ is parallel to the ...
2
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1answer
33 views

Integrate a partial derivative

If we define the operator $\mathcal{G}f(t,x)= \frac{\partial f}{\partial t}(t,x)$, what is the value of $$ \int_0^t \mathcal{G}f(s, b(s)) ds? $$ I'm sure it's some subtlety in the Fundamental Theorem ...
4
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1answer
49 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
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0answers
23 views

Multivariable calculus integration over a rectangle

Please help me. I have been stuck on this question for quite a while A parallelogram S in the xy-plane has vertices $(0,0)$, $(2, 10)$, $(3, 17)$, and $(1, 7)$. (a) Find a linear transformation $u ...
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1answer
27 views

Simple limit in multi variable II

For $x=(x_1,x_2,x_3)$ calculate (if it exists) the limit $$\lim_{x\to 0} \frac{e^{|x|^2}-1}{|x|^2+x_1^2x_2+x_2^2x_3+x_3^2x_1}$$ Solution: Let ...
2
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1answer
42 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
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1answer
26 views

Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$. $$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$ $$\frac{\partial }{\partial x} \frac{\mathrm{d} ...
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1answer
15 views

Proof: Indepence of path of integration for line integral of second type

Given the line integral of the second type: $$ I = \int_{(1,\pi)}^{(2,\pi)}\left (1-\frac{y^2}{x^2}\cos{\frac{y}{x}}\right )dx + \left (\sin{\frac{y}{x}} +\frac{y}{x}\cos{\frac{y}{x}}\right )dy $$ ...
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1answer
23 views

Application of Implicit Function Thm

Problem Let $f_{1},f_{2}$: $R^{2}\rightarrow R$ of class $C^2$. Consider the zero sets $Z_{1}, Z_{2}$ (of $f_{1},f_{2}$ respectively) ie $Z_{i}=\{(x,y) | f_{i}(x,y)=0\}$. Assume $\nabla f_{i}(x,y) ...
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0answers
9 views

Dealing with gradient of generative adversarial network

I am currently working on a recurrent implementation of something called a "Generative Adversarial Network". (link: http://arxiv.org/abs/1406.2661 ) Simply explained these are two neural networks, ...
1
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1answer
36 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
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1answer
63 views

Symbolic contour integral evaluation

Can anyone help with the evaluation of the following contour integral : $$\oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy.$$ Where the contour $C$ is given by: What I am looking for is how to split ...
1
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1answer
26 views

Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: ...
2
votes
1answer
60 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
1
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0answers
27 views

Laplacian of composition

Let $U \subset \mathbb{R}^n$ be open and $u \in C^2(U)$ with $\Delta u(y)=0$ for all $y \in U.$ Let $\phi: V \rightarrow U$ be in $C^2(U)$, too with $V \subset \mathbb{R}^n$ open. Now, I want to ...