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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Application of the implicit function theorem on an abstractly defined function

I have worked with Thomas C.T. Michaels Analysis II to study the application of the implicit function theorem and I have a pretty solid idea of how to apply it to multivariate functions but I cannot ...
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3 views

Sufficient condition for constrained extrema

When study Lagrange multiflier theorem. I try to get a sufficient conditions for constrained extremum due to the statement of Lagrange multiflier theorem, that is Given p+1 countinously ...
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1answer
13 views

Double integral (choice of) change of variables

I'm looking for a way calculate the following integral: $$\iint_D\frac{(x-y)^2(1+2y)}{(1+x+y^2} d(x,y)$$ With $D=\{(x,y)\in \mathbb{R}^2 : 0 \leq x+y^2 \leq 4 \mbox{ and } x\leq y\leq x+2\}$. what ...
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2answers
26 views

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$ where $k(x,y) = \begin{cases} y(x-1), & \text{ $0\leq y<x\leq 1$} \\[2ex] x(y-1), & ...
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1answer
24 views

How to prove that extreme values don't exist

Let $f(x,y)=(1-2x^2-y^2)xy$, how do I prove that the $f$ does not admit extreme absolutes?
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20 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
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3answers
60 views

How to create very hard problems on Lagrange Multipliers

This is a rather odd request. I only recently started studying the Lagrange Multipliers, and was given a task to create some challenging (as much as possible) problems on them and also provide ...
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1answer
38 views

Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb ...
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1answer
35 views

Find a rigorous reference that prove the following integration by parts formula in higher dimension?

My professor in the real analysis class had state the following in class but forgot to put the reference of this formula in the power point slide. The formula for integration by parts can be ...
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1answer
31 views

Where did I got wrong with this surface integral

It appears that I don't quite have surface integrals like I thought I did. The following is a problem from the back of the book (not homework because it wasn't prescribed but I'm working it to ...
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0answers
22 views

Differential of Gibbs free energy (dG) in terms of dT and dV [on hold]

The heat capacity, isothermal compressibility, and thermal expansion coefficient can be in the answer. A hint is to use the upstairs-downstairs-inside-out rule to help.
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1answer
16 views

How do I parametrise this line integral properly?

$ F (x,y,z) = ( zy + sinx , zx - 2y , yx-z ) $ is the vector field. Find the line integral of F which has curve C given as $ x = y = z^2 $ between (0,0,0) and (1,1,1) I first did this: Take $ z=t ...
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0answers
28 views

Exercise 2.36 in 'Vector Calculus, Linear Algebra and Differential Forms' (Hubbard)

Let $A$ be an n by n diagonal matrix with diagonal entries $\lambda_1$ to $\lambda_n$, and suppose that one of the diagonal entries, say $\lambda_k$, satisfies $inf_{k\neq j}|\lambda_k - \lambda_j| ...
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1answer
31 views

Analogs to vectors — *unoriented* line segments

A real vector can be thought of as an oriented line segment. Linear algebra and multivariable calculus can be taken pretty far just by considering these types of objects (obviously there are ...
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2answers
30 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
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1answer
16 views

convert from spherical co-ordinate system to cartesian

I want to convert $ρ=\cos{ϕ}$ to Cartesian system , after conversion my answer is $z=x^2+y^2+z^2$ , but its not a sphere , what have I done wrong?
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23 views

Which solution is correct?

This is a direct application of Stokes Theorem. So $\int_\Omega div F\; dv=\int_ {\partial \Omega} F.n \; dS$ $\Rightarrow 30=\int_S F.n \; dS+\int_D F.n \; dS\Rightarrow \int_S F.n ...
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0answers
28 views

Understanding notation - strange use of the del operator

I'm currently reading a paper with the following notation with the del operator which i have never encountered before: Does $\nabla _m$ just mean $\frac{\delta}{\delta \mathbf m} $ ? Furthermore, I ...
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0answers
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I am trying to find the component of b in the direction perpendicular to a. I am trying to find an alternative route to this problem. Does this work?

c being the component of b in the direction perpendicular to a. So I used the triangular law regarding vectors. I wish I could draw a picture to make it more clear. But ill try to explain... proj ...
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1answer
26 views

Line Integrals FT usage on this strange vector field: so what are the exact conditions?

I really tried thousands of things before deciding to ask here. Searched all over the internet for an answer, but failed to find it. Let's get started with the Fundamental Theorem of Line Integrals. ...
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1answer
16 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,t) | \leq \frac ...
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1answer
16 views

Having Trouble finding a simplified power series representation.

Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
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1answer
31 views

Find the gradient of $f^*(x)=\langle (\nabla f)^{-1}(x),x)\rangle-f( (\nabla f)^{-1}(x))$ for $x \in \mathbb{R^n}$

I am stuck at the following exercise which serves as a preparation for the upcoming exam: Let $U \subset \mathbb{R}^n$ be open and $f \in C^2(U, \mathbb{R})$ such that $\det Hf(x) \neq 0, \forall ...
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2answers
23 views

Multivariable calculus converting from cartesian coordinates to cylindrical coordinates

How do I convert this from cartesian coordinates to cylindrical coordinates? I am really confused. $$\int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{x^2+y^2}^4 (x) \ dz \ dy\ dx$$ I changed ...
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2answers
21 views

Conceptual explanation of integral of divergence.

$\textbf{My understanding of divergence:}$ Consider any vector field $\textbf{u}$, then $\operatorname{div}(u) = \nabla \cdot u$. More conceptually, if I place an arbitrarily small sphere around any ...
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0answers
23 views

Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
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1answer
26 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
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2answers
11 views

Parametrization of ellipse and normal vector

Parametrization of ellipse and normal vector $$F = x^2 \mathbf i + 2x\mathbf j + z^2\mathbf k \\ C: \text{ellipse} \implies 4x^2 + y^2 = 4$$ I'm trying to find the normal-vector here. I see that ...
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0answers
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Surface integral with domain that contains an infinite cone

I'm stuck on the following question: Find $\iint\limits_S {ydS}$ where $S$ is the part of the plane $z=1+y$ that lies inside the cone $z = \sqrt {2({x^2} + {y^2})} $ I tried to parametrize $x,y$ ...
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2answers
28 views

A valid method of finding limits in two variables functions?

I was wondering if in finding the limit of a two variables function (say, $F(x,y)$), I can choose the path by let $y=f(x)$, then find the limit in the same way of that in one variable functions. For ...
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4answers
320 views

Are derivatives linear maps?

I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this $$\lim_{h\rightarrow 0} ...
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0answers
24 views

Definition of differential form

Why the definition of differential form guarantee that when we do integration using differential form, it is the same as the usual Riemann integral (before we introduce the concept of differential ...
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2answers
30 views

Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$ \int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y $$ I would really appreciate some guidance on how to go ...
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1answer
19 views

Surface integral over parabolic cylinder that lies inside another cylinder

To be precise, I'm given the following: Find $\iint_K {xdS}$ over the part of parabolic cylinder $z = \frac{{{x^2}}}{2}$ that lies inside the first octant part of the cylinder $x^2+y^2=1$. In ...
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0answers
36 views

Area of triangle and uncertainty estimation

Heron's formula states that if a plane triangle has sides $a,b{\text{ and }}c$, then its area is given by $A = \sqrt {s(s - a)(s - b)(s - c)} $, where $s = \frac{1}{2} \cdot (a + b + c)$ is half the ...
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1answer
26 views

Evaluate a double integral bounded by two circles

Evaluate the integral $\iint_R y \ dR$ where $D$ is a region between the circles $x^2+y^2=2x$ and $x^2+y^2=4$ and on the first quadrant. Is my answer true? ...
2
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1answer
10 views

Continuity of the integral as a function of the domain

Let $f: \mathbb{R}^n \to \mathbb{R}$ be integrable. Let $C \subset \mathbb{R}^n$ be measurable. Is $$ r \mapsto \int_{rC} f \, \mathrm{d} \mu, $$ where $rC = \left\{ rc \: \middle| \: c \in C ...
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1answer
38 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
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1answer
21 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...
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1answer
21 views

Why $[D_{f}(x)]^{-1}$ is continuous on $\Omega$?

Let $\Omega$ be an open set in $\mathbb{R}^n,$ and $f:\Omega\rightarrow \mathbb{R}^n $ be in $C^{1}(\Omega)$. Why: If $\forall x\in\Omega$, we have det $D_{f}(x)\ne0$, then $[D_{f}(x)]^{-1}$ is a ...
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0answers
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Not getting surface integrals

I have this problem from homework: Integrate the given problem over the given surface. $H(x,y,z)=x^2 \sqrt{5-4z}$ over the parabolic dome $z = 1-x^2-y^2, x \ge 0$ I used this formula from my book ...
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3answers
71 views

Derivative with respect to $x + t$

I am reading through Princeton's lectures in analysis and I am on the 10th page of the first book on Fourier series. In analyzing the wave equation, they state that $\xi = x + t $ and $\eta = x -t$ ...
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0answers
25 views

What is the volume inside $S$, which is the surface given by the level set $\{ (x,y,z): x^2 + xy + y^2 + z^2 =1 \}$?

The solution given uses a linear algebraic argument that doesn't seem very instructive -- and may not even be correct, I think. We notice from the equation, that the surface is a quadratic form, ...
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0answers
32 views

Area of mobious strip.

I want to find area of Mobious strip. I found parameterization $\displaystyle x(s,t) = \left(1 + \frac{t}{2} \cos \frac{s}{2} \right) \cos s$ $\displaystyle y(s,t) = \left(1 + \frac{t}{2} \cos ...
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3answers
27 views

Show that for a gradient system $\bf\dot x= f(x)$, $\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}=0$ for $1 \leq i, j \leq d$ [duplicate]

The dynamical system ${\bf \dot x} = {\bf f}({\bf x})$ is called a gradient system if there exists a function $V({\bf x})$ such that $$ {\bf f}({\bf x}) = - \nabla V({\bf x}) $$ Show that if ...
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1answer
13 views

is $f$ necessarily not injective in a neighbourhood of $p$?

Let $f:\mathbb R ^3 \rightarrow \mathbb R^3$ such that $f\in C^1$. Assume there is a point $p\in \mathbb R^3$ such that $rank(Df(p))=2$ (where $Df$ is the differential of $f$) is $f$ necessarily not ...
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Find multiple integral equation solution

$\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}((v\cos\theta\sin\phi+v')^2+(v\sin\theta\sin\phi)^2+(v\cos\phi)^2)\rho r^2\sin\theta d\theta d\phi$ Can you solve this equation please I use symbolab but I ...
0
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1answer
27 views

A multivariate function with bounded partial derivatives is Lipschitz

I'm curious if I've done this correctly -- please offer suggestions/corrections if not! I'm new to working in $\Bbb R^n$ so clear insights would be appreciated. The problem: Let $f:\Bbb R^2 \to ...
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2answers
110 views

Meaning of “dS” in flux integrals?

In a general flux integral of a level surface, of the form $$\iint{\mathbf{F}\bullet d\mathbf{S}}$$ what exactly does $d\mathbf{S}$ represent? I have seen both $d\mathbf{S} = \mathbf{\hat N}dS = ...
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1answer
11 views

Hessian determinants and meanings

Suppose $f : \mathbb{R}^2 \mapsto \mathbb{R}$ is a $C^2$ map. Can the determinant of the Hessian matrix of $f$ at the same point be different for different bases of $\mathbb{R}^2$ ? What about eigen ...