0
votes
1answer
16 views

Determine $\frac{\partial}{\partial r} \int\!\!\!\!\!\!-_{B(x,r)} \frac{r}{n} u_{xx}(y)dy$

I'd like to show $\frac{\partial}{\partial r} \int\!\!\!\!\!\!-_{B(x,r)} \frac{r}{n} u_{xx}(y)dy = \int\!\!\!\!\!\!-_{\partial B(x,r)}u_{xx}dS + (\frac{1}{n}-1)\int\!\!\!\!\!\!-_{B(x,r)} u_{xx} dy$ ...
1
vote
0answers
33 views

How do I derive this recursive function?

I have a function $S_t = a*Y_t+(1-a)*S_{t-1}$ Where $a\in (0,1)\cap \mathbb{Q}$, $t$ represents a unit of time $Y_t$ is the value at a time period $t$ I am trying to find $dS_t \over dt$ I have ...
0
votes
1answer
23 views

Non zero partial derivatives in implicit function theorem?

Simple proofs of this in 3 dimensions $z(x,y)$ impose a $dz = 0$ constraint to solve for $dy/dx$. This implies $z(x, y) = \text{constant}$. Why are the partial derivatives in general in the theorem ...
1
vote
1answer
53 views

Why geometric median cannot be solved analytically

$\DeclareMathOperator*{\argmin}{argmin}$ For a given set of $m$ points $x_1,...,x_m$ with each $x_i\in \mathbb{R}^n$, the geometric median (or the weber point) is defined as $$\argmin\limits_{y \in ...
1
vote
0answers
60 views

a simple calculation

Can anyone see how (1) lead to (2)? \begin{align} ...
0
votes
3answers
26 views

Prove that $d_n$ is a Cauchy sequence in $\mathbb{R}$

Let $(x_n$) and $(y_n)$ be Cauchy sequences in $\mathbb{R}^n$ , i.e. lim$_{n,m}$ |$x_n$ − $x_m$| = $0$ and lim$_{n,m}$ |$y_n$ − $y_m$| = $0$. For each n, let $d_n = |x_n − y_n|$. Prove that $d_n$ is a ...
0
votes
1answer
17 views

Graphing the derivative at a point

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function. The function gives the surface $\{x,y,f(x,y)\}$ in $\mathbb{R}^3$. Fix a point $(a,b)\in\mathbb{R}^2$. The derivative of $f$ at $(a,b)$ is the ...
0
votes
0answers
20 views

Question concerning what a certain notation means. Its usually in calculus

$x$ln$(1 + \frac{1}{x}) = 1 +$ln$(1 + \displaystyle\sum_{i=1}^n \displaystyle\frac{a_{i}}{x^i}) + O(x^{-n-1})$ for $x \rightarrow \infty$ and $n \in \mathbb{N}$. My question is as follows:I have seen ...
0
votes
0answers
33 views

function differentiable but not continuously differentiable

Can I found an example of a function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ such that $f$ differentiable but it is not continuously differentiable and it is not invertible on a point??? Any idea? ...
0
votes
1answer
50 views

Show that this function is convex?

So I'm supposed to show that this function is convex, but I have no idea how to go about it...I've been told to use Cauchy Schwarz in order to show that the Hessian is non-negative definite, but I'm ...
2
votes
1answer
52 views

Prove that g is continuous.

Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a continuous function and define $g : \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $g(x) = |f(x)|, x \in \mathbb{R}^{n}$ Prove that g is ...
2
votes
1answer
30 views

Jacobian matrix of the inverse of a bijective function

Let $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ be a function such that $f=f(f_1,\ldots,f_n)$ and $f_i=f_i(x_1,\ldots,x_n)$. Also, $f$ is bijective and its Jacobian matrix exists. Does$f^{-1}\,$Jacobian ...
1
vote
2answers
27 views

Finding the scalar component of $\overrightarrow{PQ}$ in the direction of $\overrightarrow{PR}$

Context I am given the 3 points: $P(3,-1,3)$, $Q(1,-1,6)$, and $R(5,0,1)$ I know that $\overrightarrow{PQ} =(3-1)\hat{i}+((-1)-(-1))\hat{j}+(3-6)\hat{k} $ $=2\hat{i}+0\hat{j}-3\hat{k}$ and ...
0
votes
0answers
27 views

Solution to Laplace's Equation in different coordinate systems

Find the general solution to Laplace's equation for spherical symmetry (everything can only depend on $r$, the radius), cylindrical symmetry (everything can only depend on $s$, the radius), and planar ...
0
votes
2answers
37 views

Determine if the following set is open, closed or neither {$x\in\mathbb{R}^{2} | x_1 + x_2 = 1$} $\subset \mathbb{R}^{2}$

my textbook doesn't have solutions to most these problems, and this one is really giving me some trouble. Any help is appreciated. Determine if the following set is open, closed or neither ...
1
vote
2answers
39 views

Taking the partial derivative of $z=F(x/y)$

I have just started on chapter for partial derivative and I have a very basic question: Please go easy on me since I just started on the topic today. Let the function be: ...
1
vote
1answer
35 views

Multivar limit $\frac{6x-2y}{9x^2-y^2}$ by approach

I'm resolving the limit of $\frac{6x-2y}{9x^2-y^2}$ when $(x,y)\to(1,3)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=mx$ and $y=x^2$. In $f(x,mx)$ ...
0
votes
1answer
29 views

Multivar limit $\frac{(2x^2).y}{x^4+y^2}$

I'm resolving the limit of $\frac{(2x^2).y}{x^4+y^2}$ when $(x,y)\to(0,0)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=x^2$ and $y=0$. In $f(x,x^2)$ ...
2
votes
1answer
22 views

Sketch parametric curve

An exercise in my textbook asks to sketch the parametrical curve of the following equation: $$x=e^t\cdot\cos(t)\\y=e^t\cdot\sin(t)\\t\ge0$$ I would usually try to solve one of the equations for t and ...
1
vote
1answer
31 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
0
votes
1answer
23 views

Chain rule for partial derrivatives

Suppose $$t' = t$$ $$x' = x - vt$$ I need to prove that $$\frac{\partial{}}{\partial{x}} = \frac{\partial{}}{\partial{x'}}$$ $$\frac{\partial{}}{\partial{t}} = \frac{\partial{}}{\partial{t'}} - v ...
0
votes
1answer
22 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
1
vote
3answers
58 views

How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?

Problem. Let $f:[0,b]\times[0,b]\to\mathbb{R}$ be continuous. Prove that $$\int_0^b\left(\int_0^xf(x,y)\;dy\right)\;dx=\int_0^b\left(\int_y^bf(x,y)\;dx\right)\;dy.\tag{1}$$ My first thought was ...
0
votes
1answer
20 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
2
votes
1answer
66 views

If the partial derivatives are $0$ is a function constant?

I am trying to prove that if we have a differentiable function: $f:\mathbb{R}^2\rightarrow \mathbb{R}$, and the partial derivatives of f is 0, then f is constant on a connected set. I am using the ...
1
vote
2answers
36 views

Why is a vector function not smooth if $r'=0$

It is stated that a vector function is smooth if its derivative is continuous and nowhere zero, but I can’t find a proof or a geometric interpretation. Is this a definition or a theorem?
1
vote
2answers
21 views

Lagrange Multipliers to determine min and max

I've got this question in a book of questions I'm doing. Can someone show me step by step how to solve this? Using Lagrange Multipliers for two constraints, determine the maximum and minimum of ...
0
votes
1answer
17 views

How to find a function with two variables from two functions with one variable

I am trying to determine a function for an algorithm I wrote. The time $t$ it takes to run depends on two variables $w$ and $l$ (with $l > 0$ and $w > 0$) I measured $t$ with a fixed $w$ ...
2
votes
2answers
31 views

A Lagrange Multiplier Problem : How to deal with this case when $b< 8$

I was trying to solve the following problem of several variables calculus given in my class.I am stuck in a particular case of the problem.Please help me to solve the problem.Thnx in advance. Find ...
0
votes
0answers
22 views

Partial derivatives using chain rule

If $u = \frac{1}{y}[\phi(ax + y) + \phi(ax - y)]$, and $\phi$ is twice differentiable, show that, $$\frac{\partial^2u}{\partial x^2} = \frac{a^2}{y^2}\frac{\partial}{\partial y}\left(y^2 ...
0
votes
0answers
14 views

Finding all directional derivatives of a function involving absolute value.

I need to find all the directions in which the directional derivative exists for the function f(x,y)=|2x+y| at the point (0,0) and their values. So I used: $$ D_v(f)=\lim_{t\to 0}\frac {f(0+tu) - ...
0
votes
1answer
29 views

How to evaluate these expressions, notation

This is a very elementary question, but it has caused be a lot of confusion these last days when working on some problems. If I write: $f'(-x)$, does it mean that I have the function f, then i ...
1
vote
2answers
37 views

Derivative of a function when it is squared.

Was wondering when you are for example finding $dw/dt$ but you are given a function like $w^2(t)=r^2-2\cos(t)$, when r is some constant, how you are supposed to solve it? Are you supposed to ...
1
vote
3answers
75 views

Find convergence of series …

How do we find the convergence of $$ s_n = \sin (1!) + \frac {\sin (2!)}{1!} + \frac {\sin (3!)}{2!} + \frac {\sin (4!)}{3!} +... \frac {\sin (n+1!)}{n!} $$ I was thinking of using convergence tests ...
0
votes
0answers
14 views

Show that $f(x, y) = (|x + y| + x + y)^n$ is differentiable everywhere for all $n > 0$

Show that $f(x, y) = (|x + y| + x + y)^n$ is differentiable everywhere for all $n > 0$. In order to show that $f(x, y)$ is differentiable we have to prove that $f_x(x, y)$ and $f_y(x, y)$ exists ...
0
votes
0answers
43 views

Is this differential equation linear?

Would an equation like this be considered an ordinary linear differential equation (linear in respect to $y$)? $$\frac{d^3y}{dt^3}\dot{}\frac{d^2x}{dt^2} + \frac{d^2y}{dt^2}\dot{}x^2+\frac{dy}{dt} ...
2
votes
1answer
48 views

condition for differentiability

Lets say you have a function: $$f: \mathbb{R}^2 \rightarrow \mathbb{R^2}=((u(x,y),v(x,y)).$$ Does it follow directly from this definition: ...
2
votes
1answer
40 views

Is this an ordinary differential equation?

If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable it is said to be an ordinary differential equation (ODE). If ...
0
votes
3answers
37 views

Derivative of Unknown Function

Let's say I have a function $c(t), t \in \mathbb{R}$ and I don't know anything about it other than it is a function of $t$. If I derive said function with ... $x$ for example, what is the result? ...
0
votes
1answer
16 views

Invariance of Laplace's approximation

Suppose that $D\subset\mathbb{R}^m$ and $g(\cdot)$ is a smooth function mapping $D$ into $\mathbb{R}$ with a unique minimum at $\hat{x}$ lying in the interior of $D$. Then, the Laplace's approximation ...
1
vote
1answer
38 views

Why such a complicated counterexample to differentiable function, which has discontinuous partial derivatives

Here a counterexample is given, that a differentiable function has not necessarily continuous partial derivatives, but I asked myself why such a complicated example is given? Would simply $$ f(x) = ...
0
votes
0answers
16 views

Quantitative modeling of biological systems (a bunch of questions that I don't know how to get started on) Applied mathematics

Document for questions: http://www.physiology.arizona.edu/sites/default/files/Physiology472572_09HW1.pdf The link above leads to the problems I've been trying to figure out for the last six hours. I ...
0
votes
1answer
70 views

Smooth curve that connects two points on a sphere

I am trying to prove that there is a smooth curve that connects two points on a sphere. I want to prove this by using the Implicit Function Theorem. (I know a lot of other ways, but I want to practice ...
0
votes
0answers
46 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
0
votes
0answers
34 views

Testing Divergence Theorem using Spherical Coordinates

I'm trying to verify the divergence theorem using spherical coordinates for the vector field $\vec{F}=r^2cos^2\theta(cos\theta\hat r-sin\theta\hat\theta)$ through the top half of the unit sphere. ...
1
vote
1answer
20 views

tangent line to the graph

this is the problem, I reached pi/6 & -pi/6 each time yet the website is saying my answer is incorrect. steps i took. 1) derivative of 4t-3tant/4t+3tant 2)yeilds 4+3sec^2(x) on bottom 3) set ...
2
votes
1answer
20 views

Existence of an exponential double integral (for the probabilists: Are the $L^p$-norms of Brownian local time integrable in the space variable?)

I have encountered the following integral and, with a lot of handwaving and some identities for Gaussian integrals (see for example ...
0
votes
1answer
60 views

Second order differential equation, orthogonality

A temperature field T(x, t) is determined by the following governing equation: $$\frac 1\alpha\frac {dT}{dt} = \frac {d^2T}{dx^2}$$ (Eq 1) T(x,t) can be expressed as a form of expansion of T(x,t) = ...
2
votes
1answer
33 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
2
votes
1answer
52 views

Evaluating $\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$

I am taking a calculus exam in less than one week, and I've stumbled upon this expression. $$\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$$ Which I know to be cos(1), but ...