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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-1
votes
2answers
18 views

Lim of 2 variable function

Please help with this: Let $$f(z,b) = \begin{cases} z\sin(1/b), & b \ne 0, \\ 0, & b=0. \end{cases} $$ I'm trying to calculate the limit $f(z,b)$ for $(z,b)$ approach to $0$, in order to ...
0
votes
0answers
15 views

PDE in $2$ dimension involving the Laplace operator and mixed boundary conditions

I have a problem in resolving the problem of $$\begin{cases}-\operatorname{div} (\nabla u)=1 & \text{ in }\quad\Omega =[0,1]\times[0,1] \\ u=0 &\text{ on }\quad \partial \Omega (0,y) \\ ...
4
votes
1answer
55 views

If $f(x,y)$ is continuous in $(x_0,y_0)$, then there is a neighborhood of $(x_0,y_0)$ such that $f(x,y)>\frac12f(x_0,y_0)$

The exercise asks me to prove 2 things: 1) $f(x,y) $ is continuous in $(x_0,y_0)$, $f(x_0,y_0)>0$ then there is a neighborhood such that $f(x,y)>\frac{1}{2}f(x_0,y_0)$ My idea: $f$ is ...
0
votes
0answers
37 views

Getting a tangent line equation

Suppose that $g$ is a differentiable function of two variables, and the equation $g(x,y)=7$ defines the curve $C$ in 2-space. Also suppose that $f$ is a differentiable function of two variables and ...
0
votes
0answers
28 views

Can I use the Inverse Function Theorem in this setting?

I've been reviewing the Inverse Function Theorem, and some of its generalizations, but I have not found a version of the theorem that would let me apply the theorem for a mapping from $\mathbb R^3$ to ...
0
votes
0answers
13 views

Determining statistical correlation between XYZ points

I have a set of two 3D points, both 3D points (two points in each set) representing the same object - just in different states (State A, State B). I'd like to see if it's possible to predict the ...
0
votes
0answers
11 views

inclination of intersection of surface $z = \frac{1}{2}\sqrt{24-x^2-2y^2}$ with plane $y=2$

Just to be sure, I should take the derivative with respect to $x$, rigth?: $$z = \frac{1}{2}\sqrt{24-x^2-2y^2}$$ You can see that the function is cutted by the plane $y=2$, then we have a function ...
5
votes
1answer
61 views

Not all tangents to a plane curve are bitangents

I'm struggling on a question from a previous qualifying exam, and I don't see a clean way to do it. Let $X\subset \mathbb{R}^2$ be a connected, 1-dimensional, real analytic submanifold, not contained ...
0
votes
1answer
70 views

Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
0
votes
1answer
44 views

Partial Derivatives in Thermodynamics

I came across with the following solution from Sandler's book: $$\begin{eqnarray}\left( \frac{\partial G}{\partial T} \right)_U &=& \frac{\partial (G,U)}{\partial (T,U)} \\ &=& ...
1
vote
0answers
30 views

How to use Stokes Theorem here?

I think we have to use the Stoke's Theorem here. So let $F=(-y^3+xz)i+(yz+x^3)j+(z^2)k$. Then Curl $F=-yi+xj+3(x^2+y^2)k$. Now $\int \int_S Curl F.n dS=\int_C F.dr=$The integral we have to compute. ...
2
votes
0answers
39 views

$u=xf(xy)$, show that $xu_{xx}-yu_{xy} = 0$

I need to show that: $$xu_{xx}-yu_{xy} = 0$$ when $$u=xf(xy)$$ So, I did: $$u_x = xyf_x(xy)+f(xy) \implies $$ $$u_{xx} = xy^2f_{xx}(xy)+2yf_x(xy)$$ and $$u_{xy} = ...
1
vote
0answers
20 views

How is the line integration defined in the most general setting?

A natural generalization of Riemann-Stieltjes integration is Lebesgue integration. Some would say that the use of Lebesgue integration would be overkill when treating differentiable or continuous ...
0
votes
1answer
26 views

Given that the function is of class $C^2$ prove the following.

Let $g:\mathbb{R} \to \mathbb{R}$ be of class $C^2$. Show that $$\lim_{h \rightarrow 0} \frac{g(a+h)-2g(a) +g(a-h)}{h^2} = g''(a)$$ How should one approach such questions? There are so many things ...
2
votes
4answers
40 views

Find the equation $ax + by + cz = d$ of the plane which has equal distance to the points $A(1, 2, 3)$ and $B(4, 5, 6)$

I was just wondering if anyone has any suggestions as to how to compute this equation? Find the equation $ax + by + cz = d$ of the plane for which every point has equal distance to the points ...
0
votes
0answers
30 views

Show that the function is of class $C^1$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that $f(0,0)=0$ and $f(x,y)= \frac{xy(x^2-y^2)}{(x^2+y^2)}$, if $(x,y) \neq (0,0)$. Show that $f$ is of class $C^1(\mathbb{R}^2)$. If we use the ...
2
votes
2answers
170 views

Given 3 spheres, find the equation of the plane that touches each of the spheres on the same side..?

I have a problem I am trying to solve, but I have no idea how to solve it. If I have 3 spheres, $A(1, 2, 0), B(4, 5, 0), \text{ and } C(1, 3, 2)$ of radius 1, how would I go about finding the ...
0
votes
0answers
25 views

Isoparametric mapping from $\mathbb{R}^2$ to $\mathbb{R}^3$?

I have a reference domain (reference element) in $\mathbb{R}^2$, (r,s), which i map into a plane (physical element) in $\mathbb{R}^3$ (x,y,z) as $$x(r,s) = \sum_{i=1}^n x^{(i)} S_i(r,s) \\ y(r,s) = ...
0
votes
1answer
18 views

Multi-Variable functions( Partial derivatives involved).

We're given : $$ f(x,y) = \begin{cases} (x^{2}+y)\sin(\dfrac{1}{x^{2}+y^{2}}) & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to show that $f_x(0,0)=0$. I found ...
0
votes
1answer
18 views

Partial Derivative Problem ( Two Variable Function).

Two commodities $Q_1$ and $Q_2$ are said to be substitute commodities if an increase in the demand of either results in a decrease in the demand of the other. Let $D_1(p_1,p_2)$ and $D_2(p_1,p_2)$ be ...
1
vote
1answer
28 views

Some confusion about where vectors emanate from,

Take the plane x+y+z=0, for example. Then the vectors (0,0,0), (1,-1,0), (1,0,-1) "lie on this plane." And to find a normal to this plane, just compute the cross-product of any two independent ...
1
vote
2answers
55 views

$\cos(47^\circ)\sin(32^\circ)$ approximation by differentials

I need to approximate $\cos(47^\circ)\sin(32^\circ)$. In order to do this, I need to use differentials. So, for a function $f(x,y)$, we have: $$f(x,y)-f(x_0,y_0)\approx \frac{\partial ...
0
votes
1answer
34 views

What is nds in the surface integral of Stokes' Theorem?

I am working on an integration problem, where Stokes' theorem seems to be applicable. I have found the curl of F, and the normal vector, n, to the surface. The surface that I will integrate over is ...
1
vote
1answer
34 views

Problem that involves partial derivative of temperature function

A circular piece of metal with radius $a$ has the temperature given by the following relation: for a point $(x,y)$, the temperature $T(x,y)$ is proportional to the square of the distance of this point ...
3
votes
3answers
81 views

Intersection of $36x^2 -9y^2+4z^2+36 = 0$ with plane $x=1$, derivative at a point

The exercise asks me to find the inclination of the line tangent to the intersection of $36x^2 -9y^2+4z^2+36 = 0$ with the plane $x=1$ in the point $(1,\sqrt{12},3)$, and then say to me that I have to ...
5
votes
1answer
31 views

Relearning multivariable calculus through differential forms

While I learned multivariable calculus a few years ago, I have never felt I understand it well enough. Now I have time to go back and correct this. Since I have been through subjects like real ...
0
votes
1answer
48 views

Continuously Differentiable in $\mathbb{R^2}$

I understand the concept of continously differentiable (first derivative is continuous) in $\mathbb{R}$, however what does it mean for the RHS of: $\dfrac{d}{dt} ...
2
votes
3answers
86 views

Differentiability of this picewise function

$$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ f(x,y) = 0 & (x,y)=(0,0) \end{array}\right.$$ In order to verify if this function is differentiable, I tried to ...
4
votes
3answers
165 views

Lagrange Multipliers Example

Minimize $$f(x,y) = x^2+y^2$$ subject to the constraint $xy=3$. I know the formula for Lagrange multipliers to be $\nabla f = \lambda \nabla g$ so we get a system of equations like this ...
0
votes
1answer
12 views

Range of a function taking matrices to matrices contains a given set

Consider $M_2(\mathbb{R})$ to be the space of all real $2\times 2$ matrices. Define $$S_r:=\{A\in M_2(\mathbb{R})\lvert \max_{i,j}|a_{ij}|\leq r\}$$ for $r>0$. Now let $\Psi :M_2(\mathbb{R})\to ...
0
votes
1answer
28 views

Is Schwartz space remain closed under osilation?

Suppose that $f\in \mathcal{S}(\mathbb R^{2})$ (Schwartz space). Let $\bar{y}=(y_{1}, y_{2}) \in \mathbb R^{2}.$ Define $F:\mathbb R^{2}\to \mathbb C$ as follows: $$F(\bar{x})= ...
-2
votes
5answers
33 views

$4x + 6y =36$ represents a straight line graph. Find $x$ when $y=0$

$4x + 6y =36$ represents a straight line graph. Find i) $x$ when $y=0$ ii) $y$ when $x=0$ I don't know how to start this. I was thinking of $6y= 4x -36$ or $y=(m)(x)=0$ but it's all confusing ...
0
votes
1answer
17 views

volume of the parallepiped spanned by the vectors

Hi I am having difficulty with part (2) of the following proposition. Suppose that $x,y,z\in\mathbb{R}^3$, then (1) $\|x\times y\|=\|x\|\|y\|\sin\theta$ is the area of the parallelogram spanned by ...
0
votes
1answer
44 views

Divergence theorem to prove a relation.

I found this problem, I wanna do. Let $f$ be continuesly differentiable, let $r=\sqrt{x^2+y^2+z^2}$ Prove $\iint \limits _S \vec f (r) \cdot \vec n \ \Bbb dS = \iiint \limits _B \vec {f'(r)} \cdot ...
0
votes
1answer
39 views

change of variables in multivariate integration

let $y \in \mathbb{R}^n, s \in \mathbb{R}$ be two variables and $\theta \in S^{n-1}$ a constant. In the proof that I am reading, we have the following $\begin{eqnarray} \int_{-\infty}^{\infty} ...
3
votes
2answers
109 views

Integrating $ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$

I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?: $$ \frac{{ \int_{0}^{\infty} e^{-x^2}\, ...
1
vote
2answers
53 views

Mean value theorem for vector valued multivariable function

In the general situation of $f:S\to \mathbb R^m$ where $S\subset \mathbb R^n$. There is a form of the mean value theorem: $a\cdot (f(y)-f(x))=a\cdot (f'(z)(y-x))$ which requires a vector $a$ and dot ...
0
votes
0answers
30 views

Differentiability proof of two variable function

Show that the function is differentiable: $$f\left(x,y\right) = \begin{cases} \frac{1}{y} \sin \left( xy \right) &\mbox{if } y \ne 0 \\ x & \mbox{if } y =0. \end{cases} $$ I know the general ...
0
votes
2answers
22 views

Multivariable limit of $\sqrt{x^2+y^2} \sin (\frac{1}{x^2+y^2})$

Find Multivariable limit of $$\lim \limits_{(x,y) \rightarrow (0,0) }\sqrt{x^2+y^2} \sin \left(\frac{1}{x^2+y^2}\right)$$ Limit is obviously zero, but my question is how to simplify it a bit more to ...
5
votes
0answers
55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
3
votes
2answers
86 views

Derivative of $f(3x+1,3x-1)=4$

This exercise asks me to take the derivative of $$f(3x+1,3x-1)=4$$ where this equality is said to be valid for all $x$. The exercise specifically asks me to prove that ...
2
votes
0answers
60 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
0
votes
0answers
16 views

Classifying stationary points without the Hessian

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be $\mathcal{C}^{\infty}$ in $\mathbb{R}^n$. I can calculate the gradient which results in an expression of the form $ \nabla_{\mathbf{a}} ...
1
vote
1answer
42 views

What is the residue obtained from this integral?

Consider the following integral in two complex variables $z_1$ and $z_2$: $$\frac{1}{(2\pi i)^2}\oint_{{|z_1|=\epsilon}\atop{|z_2|=\epsilon}}dz_1 dz_2\frac{1}{z_1 ...
2
votes
2answers
140 views

$f(x,y) = e^\left({\frac{1}{x^2+y^2-1}}\right)$ continuity

for $x^2+y^2<1$ we have: $$f(x,y) = e^\left({\frac{1}{x^2+y^2-1}}\right)$$ for $x^2+y^2\ge 1$ we have: $$f(x,y) = 0$$ I'm asked in which sets these functions are continuous. That's what I did: ...
1
vote
1answer
25 views

Calculate $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^4}$

$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^4}$$ In order to calculate this limit, I did: ...
0
votes
1answer
22 views

What is a 2-norm of a multivariable complex function?

I was wondering, is there a way to specify the $2$-norm of a multivariable complex function? For example if we have a complex function: $$f = f(x_1, x_2,\cdots, x_n) = Re\{f\} + j Im\{f\}, \ x_i \in ...
0
votes
1answer
42 views

looking for the proof of a formula ($\mathbb{R}^3$ vector product) [closed]

does anyone know how to prove the following formula for vector product? $$(u\cdot\nabla)u=\nabla\frac{|u|^2}{2}-u\times(\nabla\times u).$$
0
votes
2answers
30 views

Multivariable limit of $\left(x^2+y^2 \right) \frac{1}{\sin xy}$

Find multivariable limit of: $$\lim_{(x,y)\rightarrow(0,0)}\left(x^2+y^2 \right) \frac{1}{\sin xy}$$ How to find it my attempt was to substitute $(x_n,y_n)=(\frac{1}{n}, \frac{1}{n})$ and ...
2
votes
0answers
15 views

Stuck on a limits mixed of integral and functions

$f(x,y)$ is continuous on $D=\{(x,y)\mid 0\le x,y\le1\}$ and is differentiable at point$(0,0)$ , $f(0,0)=0$, $f'_y(0,0)=1$. Calculate the following limits. ...