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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Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
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1answer
25 views

Taylor Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
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1answer
43 views

what is the difference between $f(x;y)$ and $f(x|y)$?

I was wondering whether there is a rigorous difference between $f(x;y)$ and $f(x|y)$, or if both mean the same.
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1answer
30 views

Inequality for the gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$ be a real constant. Does the ...
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1answer
49 views

Why is the Lagrange Multipliers Theorem not working?

Consider the function $h: K \to \mathbb{R}$ where $K := \{x \in \mathbb{R}^3:x,y,z \geq 0, x+2y+3z\leq 6\}$. $h$ is defined as: $$ h(x) = xe^{(x+2y+3z)} $$ Find the supremum and the ...
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2answers
82 views

Condition for equality of mixed derivatives

It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of $D_{r,r}f$ and $D_{k,k}f$? Here $f$ is a function from $\mathbb{R}^n$ to ...
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2answers
34 views

Injectivity of the function $x||x||$ on $\mathbb R^n$

Let , $f:\mathbb R^n\to \mathbb R^n$ be a function defined by $f(x)=x||x||^2$ for $x\in \mathbb R^n$. Then , which are correct ? (A) $f$ is one-one. (B) $f$ has an inverse. Here $f$ is not a ...
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1answer
43 views

Limits in multivariable function

$$\lim \limits_{(x, y) \to (0,0)} {x^3 + \sin(x^2+y^2)\over{y^4 + \sin(x^2+y^2)}}$$ I don't visualize a limited function anywhere to evaluate this limit (by the way, I have the information that this ...
3
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3answers
44 views

Why does $\lim_{x\to 0} \frac {\sin (xy)}{x} \to y $?

Let $f(x,y) = \frac{\sin (xy)}{x}$ for $x\neq 0$. How should you define $f(0,y)$ for $y\in \mathbb{R}$ so as to make $f$ a continuous function on all of $\mathbb{R}^2$? So in order for a function to ...
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1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
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0answers
49 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
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3answers
140 views

Why Lagrange multipliers don't help to find the minimum of $f(x,y)=x^2+y^2$ with the constraint $y=1$?

Please help me understand why the following doesn't work. Say I want to find the minimum of the function $f(x,y)=x^2+y^2$ with the constraint $y=1$. So I declare the helper function ...
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0answers
19 views

Absolute convergence of vector series proof

In Hubbard's multivariable calculus book there is this theorem: If $\sum_{i=1}^{\infty}|\vec a_i|$ converges, then $\sum_{i=1}^{\infty}\vec a_i$ converges. It is said in the book that the ...
3
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2answers
30 views

Graphs of functions and level sets

While going through the first few chapters of my multivariable calculus book, I came across the following: The graph of a function of two variables is a surface in $\mathbb{R}^3$ and is a level ...
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0answers
38 views

Spivak's smooth partition of unity [duplicate]

You are right for your link But In your address, There is not any solution for this question and somebody had said that $f$ is redandant without that present even a reason or one proof or a rational ...
2
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2answers
50 views

Distance and absolute value differences?

My textbook: '.. the length of a vector is in many ways analogous to the absolute value of a real number.' My question: How are the length of a vector and the absolute value of a real number ...
2
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1answer
17 views

Elementary surface integral computation

I'm working on studying for the GRE. I did this problem from Stewart's Calculus, but my answer differs from that in the back of the book. The problem is: Find the area of the part of the sphere ...
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1answer
30 views

Problem about a multivariable calculus

Decide for which of the functions $F:\mathbb R^3\to\mathbb R^3$ given below , there exists a function $f:\mathbb R^3 \to \mathbb R$ such that $(\nabla f)(x)=F(x)$. (A) ...
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3answers
48 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
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2answers
20 views

How can I show that the limit of this function under these conditions does not exist?

Show that the limit of the function, $f(x,y)=\frac{xy^2}{x^2+y^4}$, does not exist when $(x,y) \to (0,0)$. I had attempted to prove this by approaching $(0, 0)$ from $y = mx$, assuming $m = -1$ and ...
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1answer
56 views

I need help to solve this function [on hold]

given that $f(x,y,z)=xy^2-y^2+z^2$ solve $$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0 $$
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1answer
36 views

Check whether this is indeed a counterexample

Let $A,B \subset \mathbb{R}$; let $Q := A \times B$; and let $f: Q \to \mathbb{R}$ be bounded. The problem is to give a counterexample to the proposition that if the Riemann integral $\int_{Q}f$ ...
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1answer
29 views

Find multi-variable function that will make the statements true.

Let x and y denote the concentrations of two proteins encoded by the genes A and B respectively. Let f(x, y) be the rate of change of the concentration of protein A. Find a formula for f(x, y), given ...
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2answers
1k views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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0answers
5 views

How to check quasi convexity or quasi concavity using principal minor

Do you check only leading principal minors for verifying Quasi Convexity or Quasi Concavity? Does border of the bordered hessian matrix consist of first derivative of original function if there are ...
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0answers
48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [on hold]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
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0answers
19 views

What exactly is the critical point of ln(xy)

Can $\ln(xy)$ be a strictly concave function without a critical point? It seems that the graph has no critical point, therefore there doesn't seem to be maximum point. What does this imply?
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1answer
43 views

Proof of Green's identity

Can anyone explain to me how to prove Green's identity by integrating the divergence theorem? I don't understand how divergence, total derivative, and Laplace are related to each other. Why is this ...
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1answer
27 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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0answers
16 views

Understanding the meaning behind plotting a “gradient vector” on a graph containing contour lines?

A rather basic question here, please do forgive any technical errors in the question. Throughout this example consider the general function w=f(x,y) I am used to visualizing derivatives the same way ...
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3answers
25 views

Is the total differential the same as the directional derivative?

The way I understand it, the total differential and the directional derivative are both linear approximations of the change in a function at a certain point. So if I know the change in $x$ and $y$ ...
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0answers
48 views

Bounding Expected Value of a piecewise function

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
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1answer
13 views

If points cannot be added, then how can we define $\lim_{m \to \infty}(a_m+b_m)$ where $a_m$ and $b_m$ are sequences of points in $\mathbb R^n$?

I am following Hubbard's multivariable calculus book. In the beginning of the book, it says that points cannot be added but vectors can. As a rule, it doesn't make sense to add points together, ...
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1answer
43 views

Derivative exists by first principles but undefined when using chain rule

Consider the function $h$ defined by \begin{align} h(z,y)=(z^3+y^3)^{\frac{1}{3}} \end{align} Then \begin{align*} h_z(0,0)&=\lim_{t\rightarrow 0}\frac{(t^3)^{\frac{1}{3}}}{t}\\ &=1 ...
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0answers
24 views

Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
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1answer
42 views

How is the Directional Derivative a linear transform?

So I know basically what a directional derivative is and how to calculate it using the gradient vector, but I'm a bit lost on the more advanced approach of looking at it as a linear transform. I've ...
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0answers
36 views

A simple question on the Lipschitz property

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is differentiable and $L$-Lipschitz, namely $$|f(x) - f(y)|\leq L ||x-y||_2 \ \ \forall x,y\in\mathbb{R}^n~.$$ How does this imply $$||\nabla f||_2\leq L\ ?$$ ...
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1answer
2k views

Newton's method for multivariable function roots

I know how to do Newton's method to find roots for a single variable function but then I got this problem and I am unsure of how to find the roots for multivariate functions using Newton's method:
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1answer
24 views

average height of a point on an arc vs hemisphere

Why isn't the average height of a point on an arc of radius a the same as the average height on a surface of radius a. Stated another way the first problem is: Find the average height of a point on a ...
2
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1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
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1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
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1answer
37 views

Defining a function that can take one OR two arguments.

This is a two part question: 1) Let's define a recursive function as so: $$f(x,y)= \begin{cases} \hfill f(x,5) \hfill & y\le0 \\ \hfill 0 \hfill & y=1\\ \hfill x+f(x,y-1) \hfill & ...
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1answer
408 views

Finding the center mass when density is not constant or uniform throughout

I am struggling on this problem: Suppose that an object in three space has six sides and vertices at $(0, 0, 0), (4, 0, 0),(0, 6, 0), (4, 6, 0), (0, 0, 5), (4, 0, 8), (0, 6, 12),$ and $(4, 6, 15)$ ...
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3answers
763 views

How can it be proved that the geometric mean function is concave?

A function $f: \mathbb R ^n \rightarrow \mathbb R $ is said to be concave if $\forall x,y \in \mathbb{R}^n, \forall \lambda \in [0,1]$ we have $ \lambda f(x) + (1-\lambda)f(y) \le f( \lambda x + ...
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1answer
56 views

Gradient of function of matrix exponential

Suppose I have a differentiable function $\phi: \mathbb{R}^{p\times p} \mapsto \mathbb{R}$ defined as $\phi(\exp(tA))$ where $t$ is a positive scalar and $A$ is a $p\times p$ real matrix. How can I ...
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1answer
24 views

Finding the maximum on an inside an octahedron

Let $B$ be the closed domain in $\mathbb{R}^3$ defined by $|x_1|+|x_2|+|x_3|\leq 1$. Find the maximum of $F(x_1,x_2,x_3)=\sum_{i=1}^3x_i^2+\sum_{i=1}^3a_ix_i$ on $B$. Using Lagrange multiplier ...
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1answer
29 views

Can there be different values of $y_p$ for one equation?

For example, consider following example: Solution given by book is this: I solved it using different approach(as shown in the pic below) & got different answer. Is my solution wrong or ...
4
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0answers
56 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
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0answers
13 views

Characterize $|\nabla f|$ as minimal function which satisfies an upper gradient inequality

Let $f \in C^1( \mathbb R^n, \mathbb R) .$ Then one by chain rule has $$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$ I have ...