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
55 views

If a multiple integral is zero over some region, can I say the integrand is zero?

Consider the following problem Let the integral of a real function of 3 real variables $F(x,y,z)$ over some volume $V$ of $R^{3}$ vanish, $\int$$\int$$\int$$dxdydz$ $F(x,y,z)$$=0$ Now assume this ...
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2answers
42 views

Proof that the limit exists using polar coordinates

I have come to conclusion that the most efficient and thorough way to prove whether or not a limit exists in three dimensions is to use polar coordinates. $lim_{x,y \to (0,0)} \frac{x^3+y^3}{x^2+y^2}...
1
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1answer
44 views

Differentiation under integral sign- Multivariable case problem

Let $f_{\theta}(x,y)=f(x\cos \theta-y\sin \theta,x\sin\theta+y\cos\theta)$, where $f\in C^2(\Bbb{R}^2)$(Is the range necessarily $\Bbb{R}^2$? This is quite ambiguous.) a function with a bounded ...
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3answers
60 views

Surface area of the part of a sphere above a hexagon

I want to calculate the surface area of the part of a half-sphere, which lies above a regular 6-gon. (Radius $r=1$) More formally, Let $G$ be the region on the $XY$-Plane, bounded by the points $\{...
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1answer
45 views

For $f_{a,b}(x)=2^{x+a-b}-abx+4$, show that for $x_1,x_2$ close enough to $2,3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$

Let $f_{a,b}(x)=2^{x+a-b}-abx+4$. Show that for $x_1$ close enough to $2$ and $x_2$ close enough to $3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$. Hint: $f_{2,2}(2)=f_{2,2}(3)=0$. It ...
1
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1answer
34 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
1
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1answer
52 views

A differentiation with first principles question for two variables

I know this question is probably quite easy but it's been some time since I've done any sort of calculus and since a google search failed to turn up anything relevant to this specific question I ...
4
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1answer
44 views

Finding extreme value

It is given: $f\left(x,y\right)=-7x^{2}-5xy+4y^{2} $ and I should find x coordinate of the extreme value with condition $x-4y=4$. I think I know how to do this, but my solution is not the correct....
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0answers
32 views

Extremas on multivariable functions

if the gradient Of a function f(x,y,z) has all its partial derivatives (fx, fy) at a point p equal to zero but the partial derivative z at that point is equal to a constant i.e fz= 12. In this case ...
0
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1answer
86 views

Examine where $f: \Bbb R^2 \rightarrow \Bbb R$ is partial differentiable

Given, $f: \Bbb R^2 \rightarrow \Bbb R,$ $f(x, y)$ $:=$ $x^3 \over \sqrt{x^2 + y^2}$, $(x, y) \in \Bbb R^2 \setminus 0,$ $f(x, y) := 0, (x, y) = 0,$ I have to examine where the ...
2
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1answer
36 views

Examine whether the function $f(x, y)$ is continuous on $\Bbb R^2$ or not

Given, $f: \Bbb R^2 \rightarrow \Bbb R,$ $$f(x, y) := |\frac y {x^2}| e^{-|\frac y {x^2}|}, x \neq 0, y \in \Bbb R,$$ $$f(x, y) := 0, x = 0,$$ I have to decide whether the function ...
0
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0answers
16 views

Homography 4pt parameterization warp jacobian

One can parameterize a Homography matrix using 8 elements from the matrix but it can also be done using the locations of 4 points that are mapped to by 4 fixed points. I am trying to use this ...
1
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1answer
46 views

CALC III: Vector Functions and Space Curves

I'm a little confused on how to approach the question, I understand I take the derivative of the parameters, and then plug in the point. But, what do I do with the plane ? Find the points of ...
2
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1answer
39 views

Equal partial derivatives implies symmetry

The statement reads as follows: Let $f$ be continuously differentiable if $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)$ then $f(a,-a)=f(-a,a)$ I would like to know if this ...
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0answers
34 views

Center of a mass for linear $f$ over $E$ with $(x,y,z)\in E \iff (x\cos\theta-y\sin\theta,x\sin \theta+y\cos\theta,z)\in E$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
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0answers
26 views

A proof of the test of exactness for differential equations

I went through a proof of the following theorem for test of exactness of differential equations: Let the functions $M(x,y)$, $N(x,y)$, $M_y(x,y)$, and $N_x(x,y)$, be continuous on the region $R=\{(x,...
3
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2answers
92 views

Prove the following identity for the Apéry constant

Perhaps this kind of integral is well knonw, or can be easily deduced from other. I don't know it but I would like to see the computation of this to refresh the computation of iterated integrals. I ...
0
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0answers
61 views

Proof of an equation in vector calculus

How do I prove the following by integration by parts: $$\iiint \left( \vec{\triangledown} . \dfrac{\vec{I}}{r} \right)dV=\iint\dfrac{1}{r} \vec{I}.\hat{n} dS-\iiint\dfrac{1}{r} \left( \vec{\...
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1answer
28 views

Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...
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0answers
8 views

Hints to write an example of the Implicit function theorem involving $\Re\zeta(s)$ and $\Im\zeta(s)$ for $0<\Re s<1$

I would like to explore the following function, and refresh my knowledges of analysis in several real variables. I know the statement of the Implicit function theorem (previous is the Wikipedia Page), ...
1
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2answers
48 views

Double integral over semi-circular domain

Let $D=\{(x,y)\in R^2 : x^2+y^2\le1,\space x\ge0\}$, then \begin{equation} \int \int_D \frac{tan(y)}{1+x^2+y^2}dydx\ge1 \end{equation} I do not know how to approach the problem. I have tried ...
1
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1answer
32 views

Calculate Section (area) of N-Dimensional Tube

I have the following n-dimensional shape $1=\sum_{i=1}^{n}a_{i}x_{i}^{2}$ where $a_{i}>0$ and I'd like to calculate the cross-section area inside. Any suggestion? Note: I call it an $n$-...
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0answers
48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
3
votes
1answer
24 views

Is the application which associates a polynomial with its root continuous?

Let $f:\mathbb{R}^{2n+1}\to\mathbb{R}$ be defined through: $f\left(x_0,...,x_{2n}\right)$ is the greatest root of the polynomial $p(t)=\sum_{k=0}^{2n}x_kt^k$. Is $f$ continuous? If so, what is the ...
3
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2answers
41 views

“Inverse” Helmholtz Decomposition

So I am trying to write a report on the Helmholtz decomposition theorem on $\mathbb{R}^3$. The theorem states that under certain conditions, every vector field $\textbf{F}:U \subseteq \mathbb{R}^3 \to ...
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0answers
21 views

$\int_L (xe^x-x^2y)dx+(xy^2+\ln(1+y)dy$ dependency of path

Does the line integral $\int_L (xe^x-x^2y)dx+(xy^2+\ln(1+y)dy$ dependent on specific path? we were told that for $\int_Lfdx+gdy$, the integral is undependent of path if $\frac{dg}{dx}=\frac{df}{dy}$, ...
1
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0answers
81 views

Continuity of $f(x, y)=\frac{xy}{\sqrt{|x|} +y^2}$ at $(0,0)$

Assume that $f: \Bbb R^2 \rightarrow \Bbb R$ is defined by $f(0,0)=0$ and, for every $(x,y)\ne(0,0)$, $$f(x, y)={xy \over {\sqrt{|x|} + y^2}}.$$ I have to check whether the function is continuous ...
3
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1answer
56 views

Solving $\iint_{[0,1]^2}\frac{1}{(\sqrt{1+x^2+y^2})^3}$

Solving $$\iint_{[0,1]^2}\frac{1}{(\sqrt{1+x^2+y^2})^3}$$ Thinking of using $$x=r\cos\varphi \\ y=r\sin\varphi \\ J=r$$ But this is a square, so I just, am guessing need to find the range of $r$, ...
2
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1answer
51 views

Change of variable Theorem problem. What am I doing wrong?

I posted a question here the other day, asking me to show that $\int_{\Bbb{R}^n}f=nV\int_{0}^{\infty}g(r)r^{n-1}dr$ where $V$ is the volume of the unit ball (with respect to the $p(>1)$ norm.), and ...
0
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1answer
86 views

How to solve this triple integral on $[-1,1]^3$?

I need to use this in a Gaus-Ostragradski problem, but I am at a loss as to how to solve this: $$ \iiint_{[-1,1]^3}\frac{4}{(\sqrt{x^2+y^2+z^2})^3}dxdydz$$ I am told this am be done using spherical ...
0
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2answers
27 views

Derivative of function defined by integral of different variable

I have the following exercise which I certainly have gotten no clue about it. Let F(t) be defined: $F(t) = \int_{tan(t)}^{\sqrt{t^2+1}} e^{-tx^2}dx$ What is $F'(0)$? I have no clue about ...
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4answers
52 views

Why this function is not surjective?

$f: \mathbb{R^2} \rightarrow \mathbb{R^2}$ defined by $f(x,y) = ( x+y , xy )$ MY claim: By solving for (x,y) Assume $f(x,y) = ( x+y , xy ) = (a,b).$ I get $x = a +\sqrt \frac{ a^2-4b}{4}$ and $ ...
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0answers
31 views

Lagrange multipliers question with 2 constraints

Let $A=\{x\in \mathbb{R}^n|\sum x_i=n/3, \sum x_i^2=n \}$ $f(x)=\sum x_i^3$ Prove that max of f on A is of the form: $x=(a,a,.....,a,b,b...,b)$ (no need to find a or b). So with Lagrange ...
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0answers
21 views

Reversible function

I need help. For which $(r, θ, φ) ∈ \mathbb{R}^3$ is the function $$f(r,\theta,\varphi)=\begin{pmatrix}x(r,\theta,\varphi)\\ y(r,\theta,\varphi)\\z(r,\theta,\varphi)\end{pmatrix}=\begin{pmatrix}r\sin ...
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0answers
13 views

Matlab for the joint distribution

I calculate one joint distribution function, and hope to implement it in Matlab. The function is a little complex: $$F_X(x)=P\left(\frac{z}{1+y}\leq x\right)= \int _{0}^{\infty}F_{Z/Y}(x(1+y))P_Y(y)...
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1answer
21 views

Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
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0answers
41 views

Suggestion of books on Integral Calculus of Several Variables

I'd like recommendations of books on integral calculus of several variables (double integral until Gauss's theorem) that contains challenging(hard) problems. And I'd like books in languages other than ...
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3answers
31 views

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$?

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$? I would find this normal vector on point $p$ with any graphic of a function like $(-z_x,-z_y,1)$, but in this case I have no $z$ ...
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0answers
31 views

Stokes theorem on oriented curve of form $(x+y+z)dx + x^2dy+xyzdz$

This is the formulation, which is not clear to me entirely. Let $S$ be the upper unit half-sphere (this probably means the set $\{(x,y,z)|x^2+y^2+z^2=1, z\geq 0\}$) in the right half-plane plane(I ...
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0answers
17 views

Find the area of the surface of the sphere $ x^2 + y^2+z^2= a^2$ which is inside the cylinder $x^2+y^2 = ay.$

$ z = \sqrt{(a^2-x^2-y^2)};\; \frac{\partial z}{\partial x} = \frac{-x}{z}\; \frac{\partial z}{\partial y} = \frac{-y}{z};$ $\sqrt{1+\frac{x^2}{z^2}+\frac{y^2}{z^2}} = \frac{a}{z} = \frac{a}{\sqrt{a^2-...
2
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1answer
23 views

Finding the volume of Torus, Jacobian of spherical substitution.

I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\...
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0answers
66 views

Triple integral tetrahedron [on hold]

I need to find the volume of the region in the $xy$-plain bounded by: The coordinate planes The plane $x+y=4$ The plane $y+z=4$ And I am told to do so by using the triple integral in the order $...
1
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1answer
16 views

Condition for an expression to be a total differential

I have fully understood the concept and formulae around total differentials of multivariate functions. What is the condition however for an expression of differentials to be the total differential of ...
1
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2answers
28 views

Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$

I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is: $$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-...
0
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0answers
20 views

Show that $\int_{\Bbb{R}^n}f=nV\int_{0}^{\infty}g(r)r^{n-1}dr$ for $f(x)=g(||x||)$ and $g:[0,\infty)\to [0,\infty)$ integrable

Let $||\cdot||$ be the $p$-norm with $p>1$, and let $g:[0,\infty)\to[0,\infty)$. Let $f:\Bbb{R}^n\to [0,\infty)$ be defined by $f(x)=g(||x||)$. Show that if $g$ is integrable, so is $f$, and that $...
1
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1answer
49 views

Proof of a famous theorem in multivariable calculus

How do I prove the following famous theorem using simple calculus: $\int\dfrac{d}{dt}f(t,x)dx=\dfrac{d}{dt}\int f(t,x)dx$ or how is it that by first differentiating and then integrating a multi-...
1
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2answers
28 views

Find volume of cone in sphere

I can't understand why the projection of those two objects on plane $xy$ gives that the angle's range from -90 degress to 90 degrees. why not from 0 to 360? Edit: from my understanding, if $x > ...
3
votes
2answers
35 views

Conjecture about Cal 1 derivatives?

Conjecture: Let $F\left(\vec{x}\right) : \Bbb{R}^n \to \Bbb{R}$ Define $g(t) = F(t, t, \dots, t)$ Then $$g^{\prime} (t) = \left(\sum_{i=1}^n \ { \partial F \over \partial x_i}\right)\...
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0answers
15 views

Using Gauss Divergence Theorem Please help calculate

Please help me. I found div F. But I cannot handle the w.