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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convex spectral function

I am trying to understand the proof of Theorem 4.3 of S. Friedland. Convex Spectral Functions. Linear Multilinear Algebra, 9:299--316, 1981. The theorem states as follows: Let $A^{-1}$ be an ...
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1answer
188 views

Find the volume

Find the volume of the largest rectangular box in the first octant with the three faces in the coordinate planes and one vertex in the plane x+2y+3z=3. Now I know that V=xyz and have set ...
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Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
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Different Definitions of the Directional Derivative

I have seen several different starting points for definition the directional derivative of a function $f$ at a point $p$. Ultimately though, they can all be reduced to the equivalent definition via ...
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1answer
302 views

C.H. Edwards “Advanced Calculus of Several Variables”, Problem 3.5 of page 194

In C.H. Edward's Advanced Calculus of Several Variables in the Chapter III in Section 3 on Inverse and Implicit Mapping Theorems question #5 is given as follows: 3.5 Show that the equations $$ ...
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Gradient of a vector field?

What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field... thank you.
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Equation of the line passing through the origin and parallel to the planes $x+y+z=-1$ and $x-y+z=1$

Find a vector equation of the line that passes through the origin and is parallel to the planes $x+y+z=-1, x-y+z=1$ Is the answer $2x-2z=0$? I took the normals of the two planes which are $(1, 1, ...
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30 views

gradient of spectral radius

I would like to get your help with this problem. Suppose $A$ is an $n$ by $n$ irreducible nonnegative matrix and $D$ is a nonnegative diagonal matrix. Both $A$ and $D$ real. Suppose the spectral ...
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1answer
127 views

If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
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27 views

Higher Order Torsion

Define an k-Torsion as a measure of how much a parametrically defined curve $x(t)$ where $t$ is a real scalar and $x$ is a vector in $R^n$ deviates from the locally encapsulating k-dimensional ...
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1answer
58 views

Unit vector orthogonal to a surface

Not too sure about this. A surface is described as $y=\phi(x,t)$ Find a unit vector orthogonal to the surface. I was thinking of a new function $f(x,y,t) = y - \phi(x,t) = 0$ and taking the ...
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397 views

How to solve this multiple integral of Hypergeometric function?

Sorry for the typeset in the previous post, could you please help me with this integral? kind regards Ara $$ \int _0^{2\pi }d\phi \int _0^{1}du\left(u\cos^2(\phi )-1\right) \,_2F_1\left(1,\Delta ...
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5answers
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(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
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2answers
194 views

How to prove that $f(x,y)=\sqrt{x^2+y^2}$ is continuous in $\mathbb{R}^2$? [duplicate]

Please, I need the demonstration (step by step) of the continuity in $\mathbb{R}^2$ of the function $f(x,y)=\sqrt{x^2+y^2}$. I know that the function is continuous in $\mathbb{R}^2$, but I just don´t ...
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69 views

A problem on local maxima and minima

I have been asked to find the local maxima/minima of the following two functions : $$i) f(x)=0.5(x_1+x_2+x_3)^2 + 0.5(x_1-x_3)^2 + x_1-2x_2+x_3$$ $$ii) ...
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Why is positivity of $g$ required for $(x,v) \mapsto (x,g(x)v)$ to be smooth?

An exercise in Guillemin and Pollock (1.8.2) assumes that $g$ is a smooth, everywhere positive function on a manifold $X$. The book assumes all manifolds are embedding into some ambient Euclidean ...
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1answer
50 views

A basic doubt on local minima when first and second derivative test fails

How to check whether the functions $f(x,y)=x^2y^2$ and $f(x,y)=x^2y^3$ has local minima at $(0,0)$. Actually, the problem is the first and second derivative both reaches to zero at $(0,0)$. So, ...
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1answer
205 views

Is $f$ continuous at $(0,0)$

$$ f(x,y) = \begin{cases} \frac{xy^2}{x^2 + y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0)\end{cases} $$ (i) Is $f$ continuous at $(0,0)$? At $(x,y) \neq (0,0)$ this ...
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Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
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1answer
98 views

Find where the limit does not exist for the function

Given the function: $f(x,y) = \frac{xy^4}{x^2+y^8}$, find a path where the limit does not exist at the origin. I am having problems with this because of lot of paths go to $0$ but I know the limit ...
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1answer
162 views

Question about integral equations

Consider the equation $$g(t) = \int_a^b K(t,s)f(s) ds $$ where $g$ and the kernel $K$ are known and $f$ is to be determined. Suppose that the equation has a solution. Under what conditions on the ...
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1answer
35 views

Partial derivatives with two equations

I am working on this textbook problem but I am not sure how to go about this because of all the variable names confusing me...I know that if you have a function u= x+y and you want the partial ...
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2answers
197 views

Is this function continuous at the origin

I have just started learning to use the two-path test to find limits and I am very doubtful of my ability so I am verifying what I have done below is correct. I first tried letting y=0 and x -> 0 ...
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2k views

Showing limit does not exist using two-path test

I am new to using two-path test and my textbook only discusses it without showing any examples. I attempted to do this question below but I am not sure if I am correct. The question says to show the ...
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2answers
61 views

Making a multivariable Function continuous

This function $$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$ can be made continous for $f(0,y)$ by defining $$f(0, y) = 2y .$$ My question is: how can i get to this conclusion ("$2y$ must be it") ...
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1answer
95 views

multivariable function graphing

I dont at all understand how to find the domain/range of a given function. I always get these questions wrong on an exam perhaps because I lack imagination? I have a simple function below but I don't ...
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1answer
38 views

Partial derivatives in multivariable function

I was doing some practice problems and everything was going great till I saw this question out of the ordinary: $$ f(x,y) = \sum_{n=0}^\infty(xy)^n \qquad \left|xy\right| < 1 $$ Any thoughts on ...
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48 views

Solving a system of partial derivatives.

$uu_{x} - vv_{y} = 0$ $uu_{y} + vv_{x} = 0$ The subscripts represent partial derivatives. In general, the solution to this system should just be $0$. Not sure how to get that though. I was playing ...
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Find $\lim_{(x,y) \to (0,0)} \frac{\sin{(x^3 + y^5)}}{x^2 + y^4}$. Prove your result.

Find $\lim_{(x,y) \to (0,0)} \frac{\sin{(x^3 + y^5)}}{x^2 + y^4}$. Prove your result. I've attempted to apply the Squeeze Theorem as such: $\frac{-1}{x^2 + y^4} \leq \frac{\sin{(x^3 + y^5)}}{x^2 + ...
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2answers
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Jacobian matrix and Hessian matrix identity

I am trying understand the following identity in two dimensions: $$ H_{XY} = J^TH_{xy}J$$ Here $x,y$ and $X,Y$ are different coordinates for $\mathbb R^2$, the $J$ is the Jacobian matrix and the $H$ ...
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How do I prove that the dot product of a point on a level set and the gradient of the level set is 0?

The question: Let $u:\mathbb{R}^n\rightarrow \mathbb{R}$ be a differentiable function. Let $I=\{x \in \mathbb{R} \| u(x)=\bar{u}\}$ be a level set of the function. Prove that if $x \in I$ then ...
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95 views

Differentiability Theorem Question

$f(x,y) = \begin{cases} \frac{1}{2} y \log(x^2+y^2), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ You may assume that this is a continuous function. Prove that f does not satisfy ...
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1answer
78 views

Advanced Calc 2 Interchange of Limit Operations Question

Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is ...
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77 views

How to compute $\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$?

How would one compute $$\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$$ if $a,b,c>0$. I've never done an integral 3 variables before! This is from a chapter called Interchange of ...
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When did Fubini's name get applied to the theorem without measures?

Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long ...
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1answer
143 views

Evaluate integral using Stokes' theorem

Evaluate the integral $\int_C \vec{F} \cdot d\vec{r}$ with $\vec{F}$ and $C$ as given and the direction integration along $C$ being clockwise as seen by a person standing at the origin. $\vec{F}=[-z, ...
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1answer
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Question about reexpressing the dot product

Suppose that I have two arbitrary 3-dimensional vectors, $\vec{a}$ and $\vec{b}$. By the definition of the dot product, I can write $$\vec{a} \cdot \vec{b} = \left|\vec{a}\right| ...
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How to calculate hard integral?

How to calculate the integral $$ \int_D \frac {\prod_{i<j}(a_i-a_j)^2\prod_{i<j}(b_i-b_j)^2} {\prod_{i,j} (a_i+b_j)^2}\,d\lambda_{2n-1},$$ where $$D:=\{ (a_1,\dots ...
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Are derivatives defined at boundaries?

Given a differentiable function $f : [-5,5] \rightarrow \mathbb{R},$ I was under the impression that the derivative $f'$ has domain $(-5,5).$ However, according to Wikipedia ...a differentiable ...
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Parametrizations of vector-valued functions

The two vector-valued functions $r_1$ and $r_2$ below give two different motions along the same path $C$. $(a) \ $ Prove that the two functions do indeed parametrize the same curve. $(b) \ $ How is ...
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Finding the shortest distance between a point and a coplex surface

I have a surface which is $z=ax+bx^2+cxy+d$, where $a,b,c$ are coefficients and $d$ is the constant. so an arbitrary point would be $(x, y, ax + bx^2 + cxy+ d)$. there are a set of points that are not ...
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127 views

Show the the series $\sum_{n,m=1}^\infty 1/(n+m)!$ is absolutely convergent and find its sum.

Show the the series $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!}$$ is absolutely convergent and find its sum. This comes from a chapter called interchange of limit operations. I tried using the ratio ...
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228 views

2- norm of a vector in spherical and cylindrical coordinates

I was wondering how the 2-norm of a vector in cylindrical and spherical coordinates looks like? Or more general, what is the idea to derive it? Does anybody have an idea?
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209 views

Need help on Stokes Theorem in surface integral

Hello and how you doing today? I just came across a problem which need to use Stokes theorem. The problems says: Evaluate the surface integral $$ \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} $$ ...
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Properties of some operator on vectors of $\mathbb{R}^2$

Suppose that $\circ$ is an operation on $\Bbb R^2$ with the following properties: For any $\vec p,\vec q \in \mathbb{R}^2$, and $t \in \mathbb{R}$, $(t \vec p ) \circ \vec q = t(\vec p \circ \vec ...
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Volume of body given with inequalities

I need to find volume of body given with this: (using spheric coordinates) $x^2+y^2+z^2 \leq4$ $\frac{\sqrt3}{3}z\leq-\sqrt{x^2+y^2}$ I know that first inequality is inside of the sphere. But I'm ...
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1answer
66 views

How to prove that union of skew lines can be $\mathbb{R}^3$ using vector calculus

How does one prove that union of skew lines can be $\mathbb{R}^3$ using vector calculus? Space-filling curve methods are available, but i would like to know the method using vector calculus, as I ...
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1answer
46 views

approaching the border

Let $U\subset \mathbb{R}^m$ $(U\neq \mathbb{R}^m)$ open and connected. Given $b \in \partial{U}$ there is some way $\varphi:[0,1]\rightarrow \mathbb{R}^m$ with finite length such that $\varphi(t) \in ...
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2answers
73 views

Deriving a conservation law using the divergence theorem

Material scientists have discovered a new fluid property called "radost" that is carried along with a fluid as it moves from one place to the next (just like a fluid's mass or momentum). Let ...