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
6 views

Implications of zero divergence ($\nabla \cdot F$) when finding the flux

Say we are given a vector field $$F=(-x^2/2+xy,xy+y^2,-3yz-3)$$ with the property $\nabla\cdot F=0$. If we would like to find the flux through the part of the surface $x^2+y^2+2z^2=3$ that lies ...
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0answers
14 views

Explicit formula using divergence theorem

Let $A=(0,1)^k$ be the open unit cube in $\Bbb{R}^k$ and let $f\in C^1(A,\Bbb{R}^k)$. If $n$ is a unit surface normal, then by the divergence theorem, $$\int_A \text{div}f(x) dx = \int_{\partial A} ...
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0answers
6 views

What is the rank of the Jacobian of $f(x,y,z) = a_1x+a_2y+a_3z+c$

$f(x,y,z) = a_1x+a_2y+a_3z+c$ where c is an arbitrary constant I computed the Jacobian which gives me $<a_1,a_2,a_3>$ How would you go about finding the rank of a vector?
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0answers
20 views

Exercise about max and min of a 2D function with absolute value

I haven't done an exercise like this so, please, tell me if the proceeding is wrong and any kind of observations that you think can help me. Find global max and min of $$f(x,y)=|x^2-y|$$ in ...
4
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1answer
55 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
3
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1answer
31 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
2
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2answers
40 views

Finding extreme values of a variable on an intersection of a sphere and a plane

Determine the minimum and maximum value of the variable $z$ defined by the curve given by: \begin{cases} x^2+y^2+z^2=1 \\ x+2y+2z=0 \end{cases} So do I need to find a function $z=f(x,y)$ or just ...
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0answers
10 views

An optimization problem with a simplex constraint

Suppose $X^i=[0,1]$ for $i=1,2,3$. $X=\prod_i X^i$ and $\mu_i$ is a measure on $B([0,1])$ and $\mu$ is the product measure. Let $f,g,h$ be $L^2(\mu)$ integrable functions satisfying $$0\leq ...
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2answers
40 views

Need visualization advice for learning partial derivatives and calculus with more than one variable.

Okay so I just recently started learning calculus with more than one variable and whilst I'm coming to grips with many of the ideas and stuff I'm finding it difficult to visualize certain things for ...
1
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0answers
19 views

Taylor series of a twice-differentiable scalar function

I've come across this passage somewhere on wikipedia: If $f(t,x)$ is a twice-differentiable scalar function, its expansion in a Taylor series is $$df = \dfrac{\partial f}{\partial t}dt + ...
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0answers
16 views

Differential calculation in multiple variables function

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
3
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1answer
22 views

Multivariable Calculus with Tensors

I'm looking for a book at the undergraduate level on multivariable calculus (for a 2nd course of multivariable calculus) that introduces and makes use of tensors to describe higher order derivatives ...
2
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0answers
30 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
6
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3answers
476 views

Why is this map called a fold?

Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$. Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis ...
0
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1answer
7 views

Derivative of the magnitude of a parametric function

I am trying to show that $d/dt$ $|r(t)|^2 = r(t)*r'(t)$, where $r(t)= <x(t), y(t), z(t)>$ and $r(t) \neq 0$. I first tried using the fact that $|r(t)|^2 = (x(t))^2+(y(t))^2+(z(t))^2$ and then ...
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2answers
29 views

Remembering the definition of the Jacobian: any tips?

I find it impossible to remember that the Jacobian of $f: \mathbb R^n \to \mathbb R^m$ is $$ \begin{pmatrix} {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} & \dots ...
0
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1answer
20 views

What does $\|u\|_{\mathcal{C}^2(\bar{\Omega})}$ mean?

What might $$\|u\|_{\mathcal{C}^k(\bar{\Omega})}$$ mean? $u$ is a sufficiently often differentiable function $\Omega \rightarrow \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ a bounded domain. It ...
0
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0answers
10 views

Does the gradient gives a natural orientation in a manifold? [duplicate]

I want to solve the following problem: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
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0answers
32 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
1
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1answer
25 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
0
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1answer
12 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ such that, ...
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3answers
41 views

Derivation of divergence in spherical coordinates from the divergence theorem

I'm trying to find the expression of the divergence of a vector field $\vec{E}$ in spherical coordinates from the theorem : $$\iint_{S(V)}(\vec{E}.\vec{n})dS = \iiint_{V}div(\vec{E})dV$$ but if I ...
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0answers
23 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
3
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0answers
46 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
3
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1answer
43 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
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0answers
13 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
0
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2answers
44 views

Determine if z is a function of x and y. $6x-4y+2z=10$

"Determine if z is a function of x and y. $6x-4y+2z=10$. Find the formula" All i did was equate for z $$z = 5-3x+2y$$ That is the formula. And It's pretty obvious that the answers are unique but i ...
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0answers
18 views

How do you find the inverse of a multivariable function?

In 1D variable calculus, you have a nice theorem that says: Suppose $f$ is differentiable and has an inverse on $I$. Suppose $x_o \in I$ and $f'(x_0) \neq 0$. Let $y_o = f(x_o)$, then ...
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1answer
40 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
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0answers
14 views

How to go about drawing slices and projections of iterated integrals

I have no idea how to go about drawing/graphing slices parallel to an iterated integral. Specifically this one: Triple Integral: $$\large{\int_0^1 \int_y^1 \int_y^x xe^{z^2} dzdxdy}$$
2
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1answer
48 views

Application Stokes's Theorem

I am a bit unsure the way Stoke's theorem is applied in this case. Evaluate $\oint\limits_C {xydx + yzdy + zxdz} $ around the triangle with vertices $(1,0,0), (0,1,0), and (0,0,1)$, oriented ...
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0answers
14 views

Drawing slices and projections of an iterated integral.

I'm having a rough time visualizing and graphing the slices and projections of this iterated integral: $\int \limits _0 ^1 \int \limits _y ^1 \int \limits _y ^x x \mathbb e ^{z^2} \space \mathbb d x ...
2
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2answers
38 views

Find minimum value of multivariable-function

A tent with 2 rectangle shaped sides (no floor) and 2 isosceles triangles shaped gables with the volume $V$ is to be constructed. Determine the height so that the minimum amount of cloth is needed. ...
0
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1answer
10 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
0
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1answer
42 views

Compute the volume element in a differentiable manifold.

Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable manifold $ M = g^{-1}(0)$. The thing is that ...
2
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0answers
24 views

Laplace-Beltrami on a Curve

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $s(x,y)=0$? I know that if we knew the parametrization of the curve, ...
1
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1answer
20 views

Help with Lagrange multipliers on an intresting function

Hi guys I am trying to do Lagrange multipliers to figure out $\lambda$ $$F=a \log(x^2-y)+b\log(x^3-z)-\lambda (x^2-y+x^3-z -1)$$ Where a and b are constants and we have the constraint $x^2-y+x^3-z ...
3
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1answer
32 views

I need some help understanding proofs for an upside-down cycloid being the tautochrone curve. Could someone show me or point me to a simple proof?

The tautochrone curve has fascinated me since I first heard about it and I want to share it with my Calculus class as an end of the year project. I think something similar to this (Demonstrating that ...
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0answers
30 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
3
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3answers
240 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
4
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0answers
36 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
0
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1answer
30 views

Change of variable of system of ODE [on hold]

I have one problem with the change of variables of this system: \begin{cases} 2y’ + z’ –y + 2z = 0 \\ y’ + 3z’ –3y +z = 0 \end{cases} with initial values $y(0) = 1$, $z(0) = 0$ I've made this ...
2
votes
1answer
62 views

integrate this double integral by any method you can. [on hold]

I'm having trouble with this double integral: $$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
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2answers
67 views

How to find this limit $\lim\limits_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$ [on hold]

How would I find this limit? $$\lim_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$$
1
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1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
0
votes
2answers
27 views

Parametrization of an intersection cylinder ellipsoid

I'm trying to parametrize the surface given by the equations : $$\frac{x^2}{2}+\frac{y^2}{2}+z^2=1$$ and $x^2+y^2=y$. I found this function : $f:[0,1] \times [0,2\pi] \to \mathbb{R}^3$, $$(r,x) ...
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1answer
53 views

Calculate this double integral [on hold]

Recently took and exam and this was one of the questions and I wanted to check if I did it right Let $R$ be the triangular region in the ($x$,$y$)-plane with vertices $(0,0)$, $(1,0)$ and $(1,2)$. ...
1
vote
1answer
33 views

Partial derivative of function $\mid xy \mid + \sin{xy}$

I need to consulte one problem, just to control my result and see if I'm/ I'm not right: I want to find $$\frac{\partial f}{\partial x}(0,0), $$ where $f(x,y) = \mid xy \mid +\sin{xy}$ for $x,y \in ...
4
votes
2answers
82 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
2
votes
1answer
29 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...