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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2answers
48 views

Using the chain rule to find the derivative of $f(x,y) = \sin(x\sin y)$

Using the chain rule to find the derivative of $f(x,y) = \sin(x\sin y)$. I write, $\sin(x \sin y) = \sin \circ (x \sin y) = \sin \circ (p \circ (x,\sin y))$ where $p(x,y) = xy$. Now the chain rules ...
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1answer
40 views

Understanding the chain rule result and the derivative (spivak)

In spivak, we have the chain rule result: $D(g\circ f)(a) = Dg(f(a))\circ D(f(a))$ Now, I don't really understand what we are composing here, I tohught $Dg(f(a))$ is a matrix and so is $D(f(a))$ so ...
1
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1answer
21 views

Integrate 2-Form over surface

Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge ...
0
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0answers
10 views

Show that a function has bounded support

Definition in my book: A function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ has bounded support if there exists a closed interval $I$ in $\mathbb{R}^n$ such that $f(x)=0$ if $x \notin I$. Now I have ...
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2answers
26 views

Shortest distance from a point to a set of points

For our classes we use a very old book that doesn't have any examples and my professor isn't making it any any easier for me to understand. So my question is about the shortest distance from a point ...
0
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1answer
39 views

show that this function is linear

let $g$ be continuous on the unit circle such that $g(-x) = -g(x)$ define $f: \mathbb{R^2} \to \mathbb{R}$ by $f(x) = |x| g(x/\|x\|)$ if $x \not= 0$ and $0$ otherwise. Show that the function ...
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0answers
22 views

Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$

Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$. Now I have found the partial derivative of f(x,y) with respect to both x and y and got the following: ...
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0answers
22 views

Optimize $x^2+y^2+2y$ over $x\leq 1$ and $x^2+y^2\leq 2$

$$x^2+y^2+2y$$ $Z_{x}: 2x$ $Z_{y}: 2y+2$ $Z_{xx}: 2$ $Z_{yy}: 2$ D: 4 $2x=0\Rightarrow x=0$ $2y=2\Rightarrow y=-1$ so $(0,-1)$ is global minimum, how can I find the global maximum?
0
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1answer
24 views

Evaluate $\int \int_{R} (2x-3y)^2(x+y)^2 dx dy$ using substitution

Evaluate $$\int \int_{R} (2x-3y)^2(x+y)^2 dx dy$$ when $R$ is the triangle blocked by the positive X-axis, negative Y-axis and $2x-3y=4$, using $u=x+y$, $v=2x-3y$. How do I find the limits of ...
0
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1answer
16 views

The determinant of the hessian matrix

when looking for a min/max/saddle point we you the notation $D: f_{xx}\cdot f_{yy}-(f_{xy}^2)$ which is the determinant of the hessian matrix. can I use $f_{yx}$ instead of $f_{xy}$ is there ...
0
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1answer
31 views

Why do we only need a double integral to compute the centre of mass in this problem

I am confused with the following; Asked to find the centre of mass of the part of the sphere $x^2+y^2+z^2=25$ above the plane $z=4$. In the solutions, it uses that $$M=\iint_{S} \delta dS$$ where ...
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1answer
40 views

Let D be the region enclosed by the polar curve $r=1+cos\theta$

Assume D is a thin plate of constant density $ρ$. Find the coordinates of the center of mass of the plate. You should compute $m$ and $Mx$ entirely by hand; you may use Mathematica to finish off the ...
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2answers
34 views

Triple integral using polar coordinates

Let $V$ be the region in $\mathbb{R}^3$ satisfying inequalities $x^2+y^2 \le 1, 0 \le z \le 1$. Sketch $V$ and calculate: $$\displaystyle \int \int \int_{V}(x^2+y^2+z^2)\,dV$$ I wrote the ...
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0answers
23 views

Is it possible to show that this function is radially unbounded?

A function $f: \mathbb{R}^n \to \mathbb{R}$ is radially unbounded if $$\lim\limits_{\|x\| \to \infty} f(x) \to \infty$$ where $\| \|$ is the Euclidean norm Let $$f(x_1,x_2) = 3x_1^2 + 2x_1x_2 + ...
4
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1answer
35 views

Apparent violation of fundamental theorem of ODEs, how to resolve?

Consider, in the $(x, y)$-plane, the family of curves given by $y = (x - c)^3$, for the various possible values of the number $c$. Denote by $v$ the unit vector field everywhere tangent to this family ...
1
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2answers
99 views

Two variables limit question

I proved that $f(x,y)= \dfrac{xy^2}{x^2 + y^3}$ does not have limit at origin. I used two paths test; first I followed the $x$ axis, then I followed $x = \frac{1}{2}(y^2 + (y^4 - 4y^3)^{1/2})$ for ...
2
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2answers
68 views

Prove $f(x)=\|x\|$ differentiable everywhere but in $\{0\}$

I have the function $f: \mathbb R ^n \to \mathbb R$ where $f(x)=\|x\|$. I have to prove that $f$ is differentiable on $E$, where $E=\mathbb R^n \setminus \{0\} $, and show its derivative (for $x \ne 0 ...
1
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2answers
34 views

Show that f is continuous

Let $f(x,y) = \frac{2x^2y}{x^4+y^2}$, unless $x=y=0$, and $f(0,0)=0$. Define $\phi (t)=(t,at)$. Show that $\lim_{t \rightarrow 0}f(\phi(t))=0.$ Thus $f$ is continuous on any straight line through ...
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0answers
15 views

min/max/saddle $z=x^3+y^3-4xy$

find min/max/saddle points $$z=x^3+y^3-4xy$$ $z_x:3x^2-4y$ $z_y:3y^2-4x$ $z_{xx}: 6x$ $z_{yy}: 6y$ $z_{xy}: -4$ $z_x: 3x^2-4y=0\Rightarrow y=\frac{3x^2}{4}$ $z_y: 3(\frac{3x^2}{4})^2 ...
2
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1answer
31 views

Find expected value of coloured segment

Suppose picking $n$ random points from $[0,1]$ interval. For each point we colour segment of least length to it's neighboring point. If length to both neighboring points is the same then we colour to ...
1
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2answers
40 views

Show that a function is not integrable

show that $f(x,y) = \dfrac{2xy}{1+x^4+y^4}$ is not $\lambda_2$ integrable. I am given the solution, and it states: $$\int f^+ \ d \lambda_2 \geq \int_{(0,\infty)^2} f^+ d \lambda_2 = \int_0^\infty ...
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0answers
4 views

How to integrate $\int_{B_r(0)} \| {\bf u} \|_2^{2k} d{\bf u}$ where $B_r(0)$ be an $n$-dimensional ball of radius $r$ centered at $0$

Suppose we have two vectors ${\bf u}$ and ${\bf w}$ in $R^n$. Also, let $B_r(0)$ be an $n$-dimensional ball of radius $r$ centered at $0$. We want to integrated \begin{align} \int_{B_r(0)} \| {\bf u}- ...
0
votes
1answer
26 views

Line Integrals in Vector Fields

In Part A of the following, $\vec{F}$ goes from $\langle y-x,x \rangle$ to $\langle cos(t)-\sin(t),\sin(t) \rangle$ with very little explanation: I would have thought that $\vec{r}(t)=\langle ...
1
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0answers
18 views

Difference between “scalar line integral” and “line integral”

What is the difference between the phrases "scalar line integral" and "line integral"? If the phrases are equivalent, what purpose does the adjective "scalar" serve in the phrase; why is it there?
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0answers
26 views

$M=M_1\cup M_2$ is not necessarily a manifold, when $M_1\cap M_2=\emptyset$, $M_i$ a manifold

While $M_1\cap M_2=\emptyset$, $M_i$ a manifold, show $M=M_1\cup M_2$ is not necessarily a manifold. Another question, prior to this one, was to show the union is a manifold, where the conditions ...
0
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1answer
25 views

order of integrtion

find the volume of a cylinder $4cos\theta$ which intersect from above by $r^2+z^2=16$ and by $z=0$ from below So the upper plane is $z=\sqrt{16-r^2}$ and the cylinder has a radius from $0$ to ...
2
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2answers
21 views

How do i show the nth iteration term given a function and a starting point?

My task is this; The function $f:\mathbb{R} \to \mathbb{R}$ is given by $f(x) = \lambda x + k$ where $\lambda, k \in \mathbb{R}$ and $\lambda \neq 1$. Show that when we are iterating $f$ with ...
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3answers
24 views

Equality of vectors

Given $\vec{a}\cdot \vec{b} = \vec{a}\cdot \vec{c}$ Does that implies $\vec{b} =\vec{c}$ if equality holds for arbitrary $\vec{a}$ ? I think this only holds if we specify $\vec{b}$ explicitly. ...
0
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0answers
24 views

Curl by dual: need help with this example

I came across this old thread here: "... Actually, there is a generalization of curl to any dimension. If you have a vector field, you can take its dual. So, the dual of $4 i + 2 j + 3 k$ would just ...
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0answers
19 views

Discrepancy between line integral over scalar field and line integral over vector field

There is a discrepancy between the line integral over a scalar field and the line integral over a vector field that is bothering me: Say $\gamma$ is a smooth curve. If $\gamma : \mathbb R\to \mathbb ...
0
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0answers
10 views

Show that a fixed point $p$ of $f:\mathbb{R}^n\to\mathbb{R}^n$, $f\in C^1$, is isolated if $1$ isn't an eigenvalue of $df(p)$.

I'm trying to prove that if $1$ is an eigenvalue of $df(p)$, then $p$ is not isolated. We know that $$f(x)=f(p)-df(p)(x-p)+r(x)$$ then, $$f(x)=p-df(p)(x-p)+r(x)$$ Let $x$ be s.t. $x-p$ is an ...
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0answers
11 views

Gradient of softmax composed with cross-entropy

I am unsure if this is more appropriate for here or CV, but since it is mostly a question about calculus, I figured posting it here would be a reasonable idea. More specifically, I am interested in ...
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0answers
28 views

Is there a practical way to verify if a function is locally injective?

For example, it was asked to me whether $$\left\{\begin{array}{l}{ u=xy\\ v=x^2+y^2 }\end{array}\right.$$ is locally injective at $(1,1)$. I don't need the answer to this specific question, I just ...
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0answers
28 views

Finding potential for vector field and compute line integral using Maple

I am given a question to be solved on Maple which gives me a vector field $$\vec{F}(x, y, z)=\left \langle \frac{y}{1+x^2y^2}, \frac{x}{1+x^2y^2}+\frac{z}{\sqrt(1-y^2z^2)}, ...
0
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2answers
36 views

Let $f:\mathbb{R}^m\to\mathbb{R}^n$ be $C^2$ and $f(tx)=t^2f(x)$, then exisis $B:\mathbb{R}^m\times\mathbb{R}^m$ bilinear s.t. $f(x)=B(x,x)$

Let $f:\mathbb{R}^m\to\mathbb{R}^n$ be $C^2$ and $f(tx)=t^2f(x)$, for all $r\in\mathbb{R}$ and all $x\in\mathbb{R}^m$, then exisis $B:\mathbb{R}^m\times\mathbb{R}^m\to\mathbb{R}^n$ bilinear s.t. ...
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1answer
30 views

Line integral using Green's Theorem considering equation of ellipse

I am given the integral $$I=\int_{C} \frac{y}{4x^2+7y^2} dx - \frac{x}{4x^2+7y^2}dy$$ where C is the rectangle with vertices $$A=(4, 7), B=(-4, 7), C=(-4, -7), D=(4, -7)$$ oriented in the ...
0
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1answer
27 views

Volume of a sphere

Using cylindrical coordinates, find the volume of the smaller part of a sphere with a radius $r$ that intersect with a plane at a distance of $h$ from the center of the sphere I understand that ...
0
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2answers
35 views

When taking derivatives of power series, why do we shift the index up?

For example, if the series starts at n=0, and we take the derivative, the index usually then starts at n=1. This increases as we continue taking derivatives, but why do we need to do this? I get ...
1
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1answer
40 views

Evaluating these two integrals

Let A be defined as: $$A=\{(x,y,z)\in \text{unit sphere}|z\ge\cos(\alpha)\} \space0\le\alpha\le\pi$$ Take $\hat N$ as the outer unit normal vector for the sphere, $\hat n$ as the unit vector at the ...
0
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1answer
33 views

Computing $\int_C(x-y)dz$ with Stokes' theorem

Define a curve, C, as the intersection of the paraboloid $z=x^2+y^2$ and the plane $z-2x-2y=1$. Orient C counterclockwise as seen from the point $(0,0,10)$. Then calculate $$\int_C(x-y)dz$$ My ...
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0answers
9 views

If $M$ is a $k$-manifold near $x_0$, then $\phi(M)$ is a $k$-manifold near $\phi(x_0)$

$M$ is $k$-manifold around $x_0$. then $\phi(M)$ is $k-$manifold around $\phi(x_0)$ where $\phi$ is a diffeomorphism, $\phi:\Bbb{R}^n\to \Bbb{R}^n$. One definition we were given is: $M$ is a ...
0
votes
1answer
23 views

evaluate a volume in cartesian coordinates

Find the volume of $\iint_{V}\int x dV$ when V is volume between $x=0,y=0,z=0,6x+3y+z=6$ So we are only looking on the positive side of XYZ axis, and $6x+3y+z=6\Rightarrow z=6-6x-3y$ So for ...
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0answers
16 views

Find the volume of a body in the first octant.

Find the volume of the region in the first octant bounded by $y^2 = 4-x$ and $y = 2z$ using double integral. Is there a mistake in the statement of the problem? When I evaluate the integral I am ...
2
votes
1answer
35 views

Second-order derivative with respect to a function of two variables.

I have a surface defined as a radius vector in spherical coordinates: $$r = r (\theta, \psi).$$ In Cartesian coordinates, the projections are calculated as follows: $$\begin{align} r_x &= r \sin ...
1
vote
1answer
35 views

Evaluating Line Integral with Green's Theorem

I'm given a line integral $$\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy$$ where C is the circle $$x^2+y^2=8$$ oriented in the counter ...
1
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0answers
42 views

If $\,\mathbf{v}(x, y, z)=e^{yx}(y\mathbf{i}+x\mathbf{j}+z\mathbf{k})\,$ is there $\,f:\mathbb R^3\to\mathbb R\,$ s.t. $\,\mathbf{v}=\nabla f$?

A question asked me to calculate $\nabla \times \mathbf{v}$ where $\,\mathbf{v}(x, y, z) = e^{yx}\left(y\mathbf{i}+x\mathbf{j}+z\mathbf{k}\right),\,$ which I've found to be $\nabla \times \mathbf{v} = ...
1
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0answers
11 views

Prove two admissible functions sum is admissible and that an admissible multiplied by a constant is admissible.

If f and g are admissible functions on Rn and c is an element of R, show that f+g and cf are admissible. I know that to be admissible the function must be continuous on a negligible set, bounded and ...
0
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0answers
15 views

Integral surface area of ellipsoid, cylindrical coordinates?

according to legendre, the surface area of an ellipsoid is given by: $S(a,b,c)=2\pi c^2 + \frac{2\pi ab}{sin(v)}\left[\frac{c^2}{a^2} F(v,b')+\frac{a^2-c^2}{a^2} E(v,b')\right]$ where a>b>c>0, ...
2
votes
1answer
42 views

Confusing moment from definition 10.10 of PMA Rudin

Rudin states that definition of 1-surface as the same as definition 6.26. For example, in definition 10.10 we can take $D=[0,1]\cup [2,3]$ which is compact in $\mathbb{R}^1$. If $F$ is $C'$ - ...
2
votes
1answer
17 views

How do I find time derivatives?

For example, how would I simplify $ \frac {d}{dt} (q - k\sin\theta)$, where $q$ and $\theta$ are both variables and $k$ is a constant? Can I distribute $\frac {d}{dt}$?