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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14 views

An application $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $C^{1}$ such that $f(x)=0$ for $x>r$ implies the value of jacobian integral is zero

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$. Suppose that exists $r>0$ such that $f(x)=0$ if $|x|\geq r$ .Prove that exists $k>0$ such that: $\displaystyle \int_{B[0,k]}$ det$...
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1answer
40 views

Continuously differentiable function injective on convex set

Can you help me solve the following exercise: (a) Let $n\in \mathbb N$ and $G \subset \mathbb R^n$ a convex set, $f:G\to \mathbb R^n$ continuously differentiable with $$det\left(\begin{matrix} \...
0
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1answer
31 views

Difficulty setting up an iterated integral

I am trying to integrate the function $\frac{1}{\sqrt{2y-y^2}}$ over the region in the first quadrant bounded by $x^2=4-2y$. Given that this region is between bounded by an convex parabola and in the ...
4
votes
2answers
236 views

What is difference between all of these derivatives?

In calculus II we were introduced to a bunch of new derivatives: the gradient, the derivative $D=\begin{bmatrix} \partial_{x_1} \\ \partial_{x_2} \\ \vdots \\ \partial_{x_n}\end{bmatrix}$, the ...
0
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1answer
13 views

Question about terminology of Munkres's Analysis on Manifold text

The definition above are given by Munkres when he defined Euclidean manifold. One question I have about terminology is that when he says " For each $p \in M$, there is an open set $V$ of $M$ ..... ", ...
1
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1answer
20 views

2nd derivative of xy w/ respect to x?

$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in between-steps. I'm using it to prove Newtons Laws work in any frame of reference. So say two guys start from the same point and ...
0
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2answers
24 views

Application of Chain Rule for Paths

I'm a graduate student and I'm currently teaching multivariable calculus. I gave my students a question about a bug traveling along a circle of radius $200$cm in the $xy$-plane. We suppose also that ...
0
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0answers
17 views

Integral of magnetic field inside cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
0
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1answer
24 views

Finding $A$ such that $\nabla \times A = B$ for given $B$.

Let $B:U \rightarrow \mathbb{R}^3$ be a $C^\infty$ vector field, where $U = \mathbb{R}^3 \backslash \{(0,0,z):z \in \mathbb{R}\}$, defined by $$B(x,y,z)=\frac1{\rho^l} (-y,x,0)$$ where $\rho = \sqrt{...
2
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2answers
48 views

$f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
0
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1answer
35 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
2
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0answers
62 views

Is this notation common in Calculus?

Okay this is going to be quite a stupid question, but to me this seems... wrong, or at the very least not completely correct. In the material I'm reading there's a part that states that $y$ evolves ...
0
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0answers
13 views

Absolute Value Multivariate Max/Min Problem

I have an expression $$\mid1+2\lambda_1(cos(\Delta x)-1)+2\lambda_2(cos(\Delta y)-1))\mid<1$$ Where $\lambda_1=\frac{\alpha \Delta t}{(\Delta x)^2}$, $\lambda_2=\frac{\alpha \Delta t}{(\Delta y)^2}...
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0answers
16 views

Multivariable Calculus: Finding the Tangent Plane using the Normal Vector

The equation of plane is given by: $c_1(x-x_0)+c_2(y-y_0)+c_3(z-z_0)=0$ At $(x,y_0,z)$, the equation of a plane can be used to find the tangent line in respect to $x$ and to find $dz/dx$ as: $z = ...
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0answers
21 views

Proof about a given plane

So, I've been scratchin my head about this one. Consider the function $f(x,y,z)=x^{2/3}+y^{2/3}+z^{2/3}$. Let $\pi$ be a plane tangent to the level surface $c$ of the function $f$, where $C$ is a ...
1
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2answers
22 views

Eliminate the parameter given $x = \tan^{2}\theta$ and $y=\sec\theta$

$x = \tan^{2} (\theta)$ and $y = \sec (\theta)$ knowing that $\tan^{2} (\theta) = (\tan (\theta))^2 = \dfrac{\sin^{2}\theta}{\cos^{2}\theta}$ and that $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...
0
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0answers
23 views

Confusion with chain rule when proving statement about tangent plane to a point in a manifold

I'm trying to prove the following: If $f:\mathbb{R}^3 \to \mathbb{R}$ is a differentiable function, $a \in \mathbb{R}$ is a regular value of $f$ and $S=f^{-1}(a)$, then for all $p \in S$ the tangent ...
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2answers
78 views
1
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1answer
24 views

Evaluate the double integral by using polar coordinate

Determine the value of $\iint_D \sin(9x^2+4y^2)dA$ where D is the region enclosed by the ellipse $9x^2+4y^2=1$? How can i evaluate it by changing the region to polar coordinate?
4
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2answers
87 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
-1
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1answer
44 views

Find the limit of $\frac{(x-1)(y-2)}{(x-1)^2+\sin^2(y-2)}$ when $(x,y)\to(1,2)$

$$ \displaystyle\lim_{(x,y)\to(1,2)}\frac{(x-1)(y-2)}{(x-1)^2+\sin^2 {(y-2)}} $$ According to Wolfram, this limit does not exist. I know that if the limit does not exist than it should show different ...
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0answers
14 views

Invertibility of bordered Hessian

I have an optimization problem: $max f(x)$ s.t. $Ax=b$, where $x \in R^n$ and $b \in R^m$, $m \le n$. I know that $f$ is strictly quasi-concave, and that $A$ has rank $m$ (linearly independent, ...
3
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1answer
23 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
1
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2answers
27 views

Partial differentiability of $f(x, y) := {x^3 - y^3 \over x^2 + y^2}$ at $(0, 0)$

I thought this task up myself, so I'd be good to know whether my solution is correct or not. :-) Given $$f(x, y) := {x^3 - y^3 \over x^2 + y^2}$$ for $(x, y) \in \Bbb R \setminus {0},$ ...
0
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1answer
25 views

Question about calculus of variation.

What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?
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1answer
12 views

n-cubes vs simplices when defining chains

When defining chains, the standard definition is formal linear combinations of n-simplices. However in Calculus on Manifolds by Spivak, he defines chains as formal linear combinations of n-cubes. I ...
0
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1answer
24 views

Prove: $\exists (a,b)\in A | \forall (x,y)\in A : \cos (x)+\cos (xy)\leq\cos(a)+\cos(ab)$

$$ A=\left\{(x,y) | -1\leq x\leq 1, |x|\leq y, x^2-2x+y^2\leq0 \right\} $$ Prove: $\exists (a,b)\in A | \forall (x,y)\in A :$ $$ \cos (x)+\cos (xy)\leq\cos(a)+\cos(ab) $$ I don't have any lead so a ...
0
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1answer
2k views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ in the ...
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0answers
12 views

terminology help: do we say a hyperplane “embedded” in higher dimensional space?

Let $L$ be a line in $\mathbb R^2$. I have a function $f$ defined form $\mathbb R^1$ to $\mathbb R^1$ and I want to use this function $f$ on $L$ and define the set $$ S:=\{(x_1,x_2)\in\mathbb R^2,\,\, ...
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0answers
34 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$? [on hold]

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
1
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2answers
32 views

eliminate the parameters

Given: $x = \frac{1}{2} \cos(\theta)$ and $y = 2\sin(\theta)$ Part a) solving the first one for theta: 1) multiply both sides by $2$: $$2x = \cos(\theta)$$ 2) divide both sides by $\cos (\...
3
votes
1answer
92 views
+100

Integrability question with a function on a box in $\mathbb{R}^2$ (bounty added)

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
2
votes
1answer
67 views

How to show that this function $f(x,y,) <0$?

I would like to show that the function $f(x,y) = -5 x^4 + 4 y^2 - 5 y^4 + x^2 (4 - 6 y^2)$ is less than $0$ for $1 < x^2 + y^2 <2$ (at least). Upon rearranging, I obtain $f(x,y) = -5(x^2+y^2)^2 ...
0
votes
0answers
29 views

How to find a suitable function for Dulac's criteria in this example?

I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below: $$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$ I would ...
2
votes
1answer
31 views

Calculating the Stokes Theorem

I was tasked with calculating $ \oint_{L}Fdr $ for when $F=xzi-j+yk$ (vetor form) and $$L = \begin{cases}z=5(x^2+y^2)-1 & \mbox{ } \mbox{} \\z=4 & \mbox{} \mbox{} \end{cases}$$ Using: ...
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2answers
24 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
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0answers
27 views

Differentiation under the integral sign and change of variables

Let $f \in C^2 \left(\mathbb{R}^2\right)$ with a bounded support, and let $f_\phi (x,y)=f(x\cos{\phi}-y\sin{\phi},x\cos{\phi}+y\sin{\phi}))$. Show that: $$\frac{d}{d\phi}\iint_{\mathbb{R}\times(0,\...
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0answers
11 views

Direction of a gradient at maximizer on the boundary

Let $u \in C(\bar{B})$ where $B=B_1(0) \subset \mathbb{R}^n$ is the unit ball. Assume $u$ attains its maximum at $x_0 \in \partial{B}$ and $\nabla u(x_0) \neq 0$. What can we say about the direction ...
3
votes
1answer
66 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
0
votes
1answer
28 views

$f:\mathbb{R^N}\rightarrow\mathbb{R}$ Definition of Partial Derivative Using Limit or Epsilon

Can someone share the exact definition of partial derivative for a function $f:\mathbb{R^N}\rightarrow\mathbb{R}$ in both limit language and epsilon-delta language? In particular, I have hard time ...
1
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1answer
32 views

curl and stokes application

I cannot fin the flux of $$F(x,y,z)=(y^2cos(xz),x^3e^{yz},-e^{-xyz})$$ through the portion of sphere $$\Sigma = \{x^2+y^2+(z-2)^2=8, z\ge0 \}$$ I think Stokes th. must be used, so in spherical ...
1
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1answer
12 views

Square root of a $C^2$ compact-support function is Lipschitz via eigenvalues of the hessian matrix

Let $f:\mathbb R^n\longrightarrow [0,+\infty)$ be a $C^2$ function with compact support. Prove that $\sqrt f$ is $L$-Lipschitz, with $L^2\leq \frac{1}{2}\lambda(f)$, where $$\lambda(f)=\max_{x\in \...
2
votes
1answer
117 views

proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\...
3
votes
1answer
188 views
+100

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
14
votes
2answers
255 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
2
votes
0answers
20 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
1
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1answer
49 views

Jacobian and Stokes Theorem

Let $f:U \to\mathbb{R}^{n}$ a $C^{2}$ function in the open set $U \subset \mathbb{R}^{n}$. Suppose $D \subset U$ is a compact domain with boundary $\partial D$ of $C^{2}$ class. If $f(x)=0$ for all $x\...
3
votes
3answers
61 views

Prove that $\lim_{(x,y)\to(1,1)} \frac {x}{y}=1$ by epsilon delta

How can I prove that $$\lim_{(x,y)\to(1,1)} \frac {x}{y}=1$$ By epsilon delta? I am trying and I am stuck: Proof: Suppose $\epsilon >0$ we want to construct $\delta = \delta (\epsilon ) $ such ...
3
votes
2answers
104 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
votes
0answers
35 views

Riemann integrability question in $\mathbb{R}^2$

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...