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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2
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1answer
31 views

Need help converting this to Polar integral and evaluating it

I have to convert this to polar integral and evaluate it. $$\int _{-1}^0\int _{-\sqrt{1-x^2}}^0\:\frac{2}{1\:+\:\sqrt{x^2\:+\:y^2}}\:dy\:dx$$ I attempted the conversion and ended up with this $$\...
1
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1answer
31 views

Limits in the Ideal Carving Equation

I am using this source to do a mathematics research paper on the ideal carving equation in skiing, however I am having trouble understanding and explaining how these limits work: In the first ...
3
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1answer
52 views

Non-gradient vector field in $\mathbb{R}^3 - \lbrace\mathbf{0}\rbrace$ with zero curl

I'm self-studying multi-variable calculus using MIT's publicly available materials. One of the practice questions for the final exam asks that I determine the truth or falsity of the following ...
0
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1answer
32 views

Finding Desired Rate of Change that Results in Same Volume

Let's say we have an elliposoid given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$. The volume is given by $V = \frac{4}{3} \pi abc$. If at a certain moment in time $a = ...
1
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0answers
44 views

Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ exist at $(0,0).$ [duplicate]

This is a practice exam question and I have no idea how to start it. Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ ...
1
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1answer
44 views

Show that $\langle \phi(x) - \phi(y), x-y \rangle \ge c|x-y|^2$ and $|\phi(x)- \phi(y)| \ge |x-y|$.

Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a function $C^1$. Suppose there is a constant $c>0$ such that $$\langle \phi'(x)h,h \rangle \ge c |h|^2, $$ for all $h \in \mathbb{R}^n$. Show ...
1
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1answer
104 views

Parametrizing intersection of a plane and surface

I'm working on… Parametrize the curve which is the intersection of the plane $2x+4y+z=4$ with the surface $z=x^2+y^2$. I tried eliminating $z$ by plugging it into the first equation and also ...
2
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0answers
68 views

Is $f(x,y)=xy/(x^{2}+y^{2}) $ differentiable or continuous?

I'm taking a course in Analysis of several variables and the text we're following is Analysis on Manifolds - Munkres. I'm having issues to interpret properly the results I'm getting in my exercises. ...
2
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2answers
78 views

Computing $\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$

I'm trying to compute the following limits and the textbook that I'm looking at suggested the following method. $$\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$$ $$\lim_{(x,y)\to (0,0)}\frac{\sin(xy)}{...
1
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1answer
38 views

finding bounds of parametric variables

Compute the area of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ You can get it in the form $x^2+(y-1)^2=(z+1)^2$ I parametrised it as $r(u,v)=(u\cos v, u\sin v+1, u-1)$. I know that ...
0
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2answers
559 views

Partial Derivative of a outer product in Vector Calculus

I am trying to compute the partial derivative of certain vector products for calculating the stiffness matrix. So we already know that For any vector $\textbf{x}$, we have 1) The derivative of the ...
2
votes
2answers
126 views

Prove that a multivariable function has a global minimum

I'm doing an Introduction to Machine Learning course by myself using some open university coursebook and it has the following question which I've tried to solve, but to no avail: Let there be a ...
1
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1answer
41 views

Computing $\sup_{\Vert x-a\Vert\le r}{\Vert g(x-a)\Vert}$ where $g$ is a linear map.

Let $E$ and $F$ two Banach space and $g:E\rightarrow F$ be a linear map. Denote $$f(x)=g(x-a),$$ I would like to know if $f$ is tangent to $0$ at $a$ i.e. $$\lim_{r\rightarrow0}\frac{m(r)}{...
0
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1answer
49 views

How to prove that multivariable function has a minima?

Consider for example the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ as: $$f\left(x,y,z\right)\:=\:z^2+3x^2+y^2-2xy+14$$ I need to prove the $f$ has a minima and to calculate it. So far i went ...
1
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1answer
78 views

Is this function continuous? Polar coordinates

Is the function $f:\mathbb R^2\to \mathbb R^2$ given in polar coordinates by $f(r,\theta)=(1,\theta)$ continuous? How would one prove it? My guess would be yes, since geometricly it simply change ...
0
votes
1answer
42 views

Find potential function for the vector field $\vec F(x)=\left \| x \right \|^px$

Define a vector field $\vec F$ in $\Bbb{R}^n \setminus 0$ by $\vec F(x)=\left \| x \right \|^px$, where $p$ is a real constant. How to find a potential function for $\vec F$? Shall I just directly ...
1
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1answer
49 views

Derivative not defined, but does exist?

Consider the function f: $\ (x,y) = y \sqrt{x^2 + y^2} $. The derivative at the origin is zero. However, if I calculate the partial derivative with respect to x, for example, I get the following: $ \...
0
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1answer
18 views

show that $c′(5)$ is orthogonal to $\nabla f(1,4,2).$

I need some help here. Let $f(x, y, z)$ be a differentiable function and suppose that $c(t)$ is a path which lies on the surface $f(x, y, z) = 17.$ If $c(5) = <1, 4, 2>$ show that $c′(5)$ is ...
2
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2answers
37 views

Trying to find function that defines a parabolic surface

Say we are working in three dimensions and we have a function $u_1(x, y) = x^2$. Ie. this is just the regular $x^2$ parabola except it's now defined all along the way $y$-axis and forms a surface. ...
1
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2answers
471 views

Using Lagrange multipliers to find the shortest distance between two straight lines

A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$ (It also warns me that using this method is a bit ...
0
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1answer
255 views

Finding the slope at a given point on the intersection of a surface and a plane

The surface with equation $z = x^{3}+xy^{2}$ intersects the plane with equation $2x−2y = 1$ in a curve. What is the slope of that curve when $x = 1$ and $y = -\frac{1}{2}$? I'm a little confused ...
1
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1answer
58 views

surface integrals parametrising

Find a parameterisation and compute $r_{\alpha},r_{\beta},r_{\alpha}$ x $r_{\beta}$ and the tangent plane at the point mentioned of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ and the ...
4
votes
2answers
74 views

Show that $f:\mathbb{R}^2\to\mathbb{R}$ is constant.

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $\left|f(x)-f(y)\right| \le \|x-y\|^2$ for every $x,y\in\mathbb{R}^2$. Show that $f$ is constant. So I think that if $f$ is constant then $\nabla f \equiv ...
2
votes
2answers
257 views

Limit of a function of two variables

This is a problem occurs to me when I was trying to find the limit of a function with two variables(if only it exists). Please help. When I have to show that the limit does not exist for some ...
1
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1answer
61 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
1
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3answers
32 views

Global max/min of surfaces

Given $f(x,y)=4x^3+4x^2y+3y^2$ and restrictions $x,y≥0$ and $x+y≤1$, I'm trying to find global max and mins. I found the partial derivative and found the critical point $(0,0)$ by setting those to $0$...
0
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2answers
53 views

critical points of the function

Find the critical points of the function $$f(x,y)=(4x-x^2)\cos y$$ first let's determinate Partial derivatives: $$\dfrac{\partial f}{\partial x}(x,y)=(4-2x)\cos y$$ $$\dfrac{\partial f}{\partial y}(...
0
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1answer
33 views

Continunity of a two variable function in Apostol's analysis

In discussing how the concept of differentiability implying continuity cannot be applied to functions of several variables, Apostol proceeds to give an example to demonstrate why. The function he uses ...
1
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1answer
49 views

prove that $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$ [duplicate]

How to prove that $f: \mathbb R^2 \to \mathbb R$ $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$? By definition: Let $\epsilon>0$ I need to prove ...
2
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4answers
161 views

Prove $f(x,y) = \frac{x^2+y^2}{x+y}$ is not continuous at $(0,0)$.

Let $f(x,y) = \frac{x^2+y^2}{x+y}$ when $x+y \neq 0$ and $f(x,y) = 0$ when $x+y=0$. Prove $f$ is not continuous at $(0,0)$ in the $R^2$ norm. Is this as easy as noticing that the function is ...
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2answers
63 views

Evaluate the following spherical coordinate triple integral

Evaluate the following \begin{align*} \int_{0}^{\pi} \int_{0}^{\pi/3} \int_{\sec \phi}^{2} 5\rho^2 \sin(\phi) \ d\rho \ d\phi \ d\theta \end{align*} Attempt at solution: We have \begin{align*} 5 \...
3
votes
1answer
154 views

Find $\lim \limits_{(x,y) \rightarrow (0,0)}\frac{x^3 y^3 }{x^2+y^2}$ (epsilon delta proof)

Suppose we have the following function: $f(x,y)=\dfrac{x^3 y^3 }{x^2+y^2}$. Determine $\lim \limits_{(x,y) \rightarrow (0,0)}f(x,y)$. I did the following, but I cannot continue. Suppose $0 < \...
0
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2answers
48 views

Integration Under the integral sign on indefinite integrals

Is it possible to perform the differentiation under the integral sign for an indefinite integral (anti-derivative)? that is, if $f(s) = \int F(s,t) dt $ then, is $f'(s) = \int (d/ds(F(s,t)))dt$ ...
1
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2answers
180 views

Can wolframAlpha be wrong on this vector limit?

We had a homework on multivariable analysis, and there was this problem and the teacher said that we didnt trust wolfram but I'm not convinced on it, because of this. Is $f(x,y)=\frac{x^2}{x^2+y^2-x}$...
0
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2answers
23 views

Finding extrema of $z(x,y)$ given some conditions

I am given the following exercise and I am not sure I understand what it asks: Find the extrema for $z=2x^2-y^2$, given $x+y=2$. This part "given $x+y=2$" is confusing. I know that if any point ...
-1
votes
1answer
27 views

Prove the following lemma

If $f$ is differentiable at $X_o$, then $f(X)-f(X_o)= (d_{x_o} f)(X-X_o)+ E(X)|X-X_o|$, where E is defined in the neighborhood of $X_o$ and $\lim_{X\to X_o}$ $E(X)=E(X_o)=0$ I don't know how to go ...
0
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1answer
22 views

What is the derivative of $|Du|$?

Let $u:\mathbb{R}^n\to \mathbb{R}$. Suppose $u$ is differentiable. What is the derivative of $|Du|$? Is it equal to $\text{div}(Du)$? Here div means the divergence. Thank you very much.
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2answers
120 views

Prove that the limit of the following function of two variables is zero

I need to prove the following: $$\lim_{(x,y)\to (1 ,2)} \frac{x^2+2xy-6x-2y+5}{\sqrt{(x-1)^2+(y-2)^2}}=0$$ I've tried to solve it by substituting $y=mx$ but I can't get the solution that way. ...
0
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2answers
55 views

How to argue that $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$ has a global maximum?

Let $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$. I know $f$ has a local maximum at $(0,0,0)$ but how do I argue that this is also the global maximum. The solution provided simply states it is a global ...
2
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0answers
54 views

Multiple Integral Puzzle [duplicate]

$$I=\int_1^2\int_1^2\int_1^2\int_1^2 {{x_1+x_2+x_3-x_4}\over{x_1+x_2+x_3+x_4}} \,dx_1\,dx_2\,dx_3\,dx_4$$ $I=1/2$ or $1/3$ or $1/4$ or $1$ ? [from ISI-Kolkatta Sample papers] I know that there ...
0
votes
1answer
29 views

Find all real values of $k$ for which the given integral converges.

In the following, $R_k$ is the region $1 \leq x \leq \infty$, $0 \leq y \leq x^k$. $b$ is just a given real number. Find all real values of $k$ for which the given integral converges: \begin{align*} \...
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0answers
75 views

Corollary of the inverse function theorem

Let $U\subset \mathbb{R}^{n}$ and $ f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...
0
votes
1answer
48 views

What does it mean to “admit” something in vector calculus?

Trying to understand the Helmholtz decomposition has lead me to the concept of a vector potential. From Wikipedia [1]: If a vector field v admits a vector potential A, then [...] I've searched ...
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0answers
38 views

$\lim(x^2+y^2)/(x-y)$ when $(x,y)\to(0,0)$ [duplicate]

How can I prove (without polar coordinates) that the $\lim \frac{x^2+y^2}{x-y}$ when $(x,y)\rightarrow (0,0)$ does not exist? Moreover, can I use $(x^2+y^2)=(x+y)(x-y)$ in this case? If I do it seems ...
1
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0answers
20 views

Finding the Local Maximum and Minimum?

I am given $f(x,y) = 6y - y^3 - 4x^2y$ and asked to find the critical points. After I have found those I'm asked to find D= $f_{xx}f_{yy} - f_{xy}^2$ and to decide whether I have a local max, min, or ...
1
vote
1answer
43 views

What is the $k$th differential of a composite map?

Let $V$, $W$, and $X$ be normed linear spaces, and let $A$ be an open subset of $V$. Suppose $F \in C^k(A,W)$ and $G \in C^k(W,X)$. What is $d^k(G\circ F)_\alpha$ (the $k$th differential of $G \circ F$...
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0answers
83 views

difficult example of a not differentiable function $f: \mathbb R^2 \to \mathbb R^2$ [closed]

Give an example of a function $f: \mathbb R^2 \to \mathbb R^2$ so that: 1) all its directional derivatives exist at $(0,0)$ ($D_{\vec u}f(0,0)$ exist for all $\vec u \in \mathbb R^2$ unitary), 2) $...
1
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0answers
52 views

I need help with divergence and gradient?

$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$ Midway into my question, I want to compute: $$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$ Symbols like $ w, \...
0
votes
1answer
44 views

Prove a consequence of the multivariable version of the inverse function theorem

The exercise is the following: Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ that is class $C^{1}$ such that there exists $c >0$ such that $$|f(x) - f(y)| \ge c|x-y|$$ for all $x,y \in \mathbb{R}^{...
0
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0answers
87 views

If $f_{xy}$ , $f_{yx}$ are continuous at $(x_{0},y_{0})$,then $f_{x},f_{y}$ are continuous at $(x_{0},y_{0})$?

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ The second edition Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume ...