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
35 views

Triple integral question- two balls in $\mathbb{R}^3$

How can I calculate the volume bounded between the two surfaces: $$ x^2+y^2+z^2=1, \quad x^2+(y-1)^2+z^2=1 $$ and contains the point $(0,0.5,0)$. When I move to spherical coordinates, I obtain ...
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2answers
34 views

tangent to a level surface

Let $F:\mathbb{R}\to \mathbb{R}^n$ be differentiable. Let $f:\mathbb{R}^n \to \mathbb{R}$ be continuously differentiable and such that the composition $g(t)=f(F(t))$ exists. If $F'(t_0)$ is tangent ...
2
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1answer
54 views

Prove that $\lim_{\epsilon \rightarrow 0}\int_{\partial B_\epsilon} (φ∇g · n − g∇φ · n) ds = 2πφ(0, 0)$

Suppose $φ : \mathbb{R^2}\rightarrow \mathbb{R}$ is any $C^1$ function and let $g:\mathbb{R^2}-\{(0,0)\}\rightarrow \mathbb{R}$ given by $g(x, y) := \ln\sqrt{x^2+y^2}$ Prove that $\lim_{\...
2
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1answer
32 views

Have I made a mistake in the resolution?(surface integral)

Problem: Compute the surface area of that portion of the sphere $x² + y² + z² = a²$ lying within the cylinder $x² + y² = ay$ , where a > 0. My Try: We consider $r(x,y)=(x,y,f(x,y))=(x,y,\sqrt{a²-x²...
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2answers
28 views

Find the limit of the vector function

$lim_{t\to\infty} \Big(te^{-t},\frac{t^3+t}{2t^3-1},tsin(\frac{1}{t})\Big)$ a) $lim_{t\to\infty} te^{-t} = \infty \times 0$ $lim_{t\to\infty} 1e^{-t}+-e^tt = 0+(0\times\infty)$=undefined, and ...
0
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1answer
21 views

Find the volume of solid that lies under $z=x^2+y^2$, the $xy$ plane and inside $x^2+y^2=2x$

Find the volume of solid that lies under $z=x^2+y^2$, the $xy$ plane and inside $x^2+y^2=2x$ My attempt: $z = x^2+y^2 = r^2= 2x$, and $x=rcos(\theta)$, so $r^2=2rcos(\theta),$ $r = 2cos(\theta)$ $...
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1answer
24 views

Curve of intersection, value for parameter

This is for a line integral. Parametrize the curve of intersection: \begin{align*} S_1: x^2+4y^2 + z^2 &= 4a^2, y<0\\ S_2: x+2y &= 0 \end{align*} Orientation from $(0,0,-2a)$ to $(0,0,2a)$....
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2answers
63 views

At any $θ$ on curve $x=a\cos{θ}+aθ \sin{θ}$,$ y=a\sin{θ}-aθ\cos{θ}$ what is distance from the origin to nomal?

The parametric equation at any point $\theta$ of a curve is: $$x=a\cos{(\theta)}+a\theta\sin{(\theta)}, y=a\sin(\theta)-a\theta\cos{(\theta)}$$ What is its distance from the origin to its normal at ...
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1answer
40 views

How is the following derivative equal to the limit on the right side of the equation?

I am puzzled on how the following derivative is equal to the limit on the right side of the equation. I have tried to use the limit definition of a derivative to explain it, but I believe I am making ...
1
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1answer
72 views

Why does the gradient of matrix product $AB$ w.r.t. $A$ equal $B^T$?

The below passage is from p. 215 of Deep Learning by Goodfellow, Bengio and Courville. For example, we might use a matrix multiplication operation to create a variable $C = AB$. Suppose that the ...
1
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1answer
30 views

Find the volume common to $r^2+z^2=a^2$ and $r=a\sin(\theta)$.

Sorry to ask another one of these, but I am really struggling with these integrals. The question asks to find the volume common to $r^2+z^2=a^2$ and $r=a\sin(\theta)$. I attempted to set up the ...
1
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1answer
18 views

Hint needed in volume integral computation: Finding the volume in the first octant inside of $y^2+z^2=16$ but outside $y^2=3x$

I am looking for a hint about how to set up the bounds incorporating the information "outside of $y^2=3x$. I know that this should look like a cylinder along the $x$ axis, the parabola $y^2=3x$ ...
2
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1answer
40 views

Bounds of third integral

For integrals: $\iiint_D x^{2}yz \, \mathrm dx\mathrm dy\mathrm dz$, where $D$ is limited by surfaces: $x = 2, y=x^2, z=0, x+y=z$. $\iiint_D xz \, \mathrm dx\mathrm dy\mathrm dz$, where $D$ is ...
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2answers
31 views

evaluation of double integral using change of order of integration

How to evaluate the following double integral $\int\limits_s^t\int\limits_s^u e^{-\lambda(t-v)}(u-v)^{-\beta-1}dvdu $ where $\lambda$ and $\beta$ are positve constants.
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0answers
31 views

calculate area using stokes

Let $S:= \{(x,y,z) | z = -x-y \} \cap \{(x,y,z) | z^2+y^2+x^2 \leq 1\}$ I have to calculate the Area of $S$. Since $S$ is the unit circle. I know that the area is $\pi$ But I have to use Stokes. I ...
3
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0answers
49 views

Integral of determinant of Jacobian of a $C^{1}$ application.

Let $f \in C^{1}(\mathbb{R}^{n}; \mathbb{R}^{n})$, suppose that exists $r>0$ such that $f(x)=0$ if $|x|>r$. Prove that exists $k>0$ such that $\int_{\overline{B_{k}(0)}} det Jf(x) dx=0$. My ...
9
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1answer
106 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
39 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 ...
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1answer
15 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 $M$ ..... ", ...
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1answer
26 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
31 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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1answer
28 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{...
3
votes
2answers
92 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 ...
2
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1answer
94 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
38 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 ...
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0answers
26 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 ...
2
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0answers
99 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 ...
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0answers
20 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
22 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 ...
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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 $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...
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2answers
82 views

How to find the limit of $\lim_{(x, y)\to (0,0)} \frac{x^2y}{x^3+y}$ [closed]

I know the limit is 0 but I can't find how show that
1
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1answer
26 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?
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0answers
26 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 ...
1
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1answer
55 views

Invertibility of bordered Hessian

I have an optimization problem: $max_{x \in C} f(x)$ s.t. $Ax=b$, where $x \in R^n$ and $b \in R^m$, $m \le n$, adn $C$ compact. I know that $f$ is strictly quasi-concave, and that $A$ has rank $m$ (...
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2answers
31 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
13 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 ...
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1answer
46 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 ...
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 ...
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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,\,\, ...
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 (\...
2
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1answer
70 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 ...
2
votes
1answer
35 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: ...
0
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0answers
31 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
0answers
36 views

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

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
4
votes
2answers
94 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 ...
0
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1answer
29 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 ...
3
votes
1answer
25 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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1answer
33 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 ...
0
votes
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 ...
0
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2answers
26 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.