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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22 views

Integration with cylindrical coordinates

I need to use cylindrical coordinates to find the volume of a region which when projected onto the xy plane is the disk $x^2+y^2+2y-3=0$. I already know what I'm integrating between for $z$ but I need ...
2
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1answer
32 views

Why do we need an open set to define differentiability? [duplicate]

The general definition of a differentiable mapping is, Let U be an open set in Rn, and let ‘a’ be in U and f:Rp. Then f is a differentiable mapping at ‘a’ if there exists a Df(a) in Hom(Rn, Rp) such ...
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1answer
27 views

How to evaluate this line integral [closed]

C is circle centered at a,0 and having radius a . wat i ve done is that x=a + acost y=a sint then wen i take bounds of theta from -pi/2 to +pi/2 gives wrong answr .why so/
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1answer
24 views

Definition of a functions with respect to partials

I am stuck with the following problem: I am given that $$F(x,y)=f(x,y,g(x,y)) =0.$$ I am asked to show $D_1g$ and $D_2g$ with respect to the partials of $f$ My idea was to write that $DF=DfDg$ ...
2
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2answers
23 views

Calculating marginal probability density when multivariate pdf's support is $0<y<2$ and $y<x<3$

Suppose that multivariate pdf $f(x,y)$'s support is in $0<y<2$ and $y<x<3$. I now want to calculate marginal probability density function $f_X(x)$ and $f_Y(y)$. But arranging terms only ...
1
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1answer
24 views

Property of homogeneous functions in two variables

Why is $f_x(tx, ty) = t^{n-1}f_x(x, y)$ when $f(x, y)$ is a homogeneous function of degree $n$? What I came up with is that if $u = tx$, because $f(tx, ty) = t^{n}f(x, y)$, $$t^n\frac{\partial ...
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1answer
27 views

Positive numbers and functions satisfying some conditions

I need to show that there are positive numbers p and q, and unique functions u and v mapping from interval $(-1-p,-1+p)$ to $(1-q,1+q)$ such that $$xe^{u(x)}+u(x)e^{v(x)}=0=xe^{v(x)}+v(x)e^{u(x)}$$ ...
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1answer
21 views

Applying the Implicit Function Theorem to the Unit Sphere

The unit sphere S given by $x^2+y^2+z^2=1$ intersects each of the three axis at 2 points, at these points, what variables can be solved for? For example, S intersects the x-axis at $(\pm1,0,0)$, I can ...
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0answers
15 views

Calculation of Jacobian

I aim to solve the following system. $x = (a(x,y))^2+3\sin b(x,y),$ $y = 2e^{a(x,y)}-\cos a(x,y)b(x,y).$ I think I should use the implicit function theorem, but I'm a bit shaky with this one ...
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1answer
16 views

Find equations of the tangent plane and the normal line to the given surface

Find equations of the tangent plane and the normal line to the given surface at the specified point $(0, 0, 6)$: $$x + y + z = 6e^{xyz}.$$
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1answer
18 views

Tangent space and implicit function theorem

Let's say we have a $C^1$-function $f:X\to\mathbb{R}^m$ ($X\subset\mathbb{R}^{n+m}$ an open set) and the rank of the matrix $Df(x)$ is $m.$ We'll let $Z=\lbrace x\in X:f(x)=0\rbrace$ and take some ...
5
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1answer
29 views

Multiple Integration order doesn't agree.

Let $0<x,y,t,z<1$ with the additional condition: $$\begin{align*} x &< t\\ \wedge & \ \\ y &<z \end{align*}$$ Call the set of all $x,y,t,z$ satisfying the above conditions ...
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0answers
15 views

Weierstrass Theorem

If $f:A\to\Bbb{R}$ is continuous in the rectangle $A=\{(x,y)\in\Bbb{R}^2|\alpha\leq x\leq\beta;\alpha'\leq y\leq\beta'\}$ is possible to show that $f$ is bounded in $A$, i.e., there is some $M>0$ ...
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2answers
22 views

Let $f(x,y) = \frac {-1}4 (3xy^2 - 5x^3y + 2x^4)$. Find the equation of the tangent plane to $f$ at the point $(2,4)$.

Let $f(x,y) = \frac {-1}4 (3xy^2 - 5x^3y + 2x^4)$. Find the equation of the tangent plane to $f$ at the point $(2,4)$. Using vector dot product with: $a = 2$ $b = 4$ $f(a,b) = -8$ $\frac ...
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3answers
29 views

What are the critical points of this multivariable function?

What are the critical points of the function $z=x^3+y^3 -12yx$? I had $(0,0)$ and $(4,4)$ as the only ones, but saw other answers in the class. Also, are the values maxes or mins, as I had $(0,0)$ was ...
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3answers
41 views

Differentiability at $(0,0)$.

I always get stuck when I've to show something is differentiable ,like in the following question: $$f(x,y) = \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & \text{if $(x,y)\neq(0,0)$} \\ 0 & ...
1
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1answer
27 views

Stokes theorem problem $\displaystyle \int_C (3y+z)dx+(x^2 +2yz)dy+(2x+y^2)dz$

Let $ S_1=\{(x,y,z) \ | \ x^2+y^2-2x-2y+1=0 \} $ $ S_2=\{(x,y,z) \ | \ 2x+3y+z=9 \} $ and $C=S_1\cap S_2$ I'd like to calculate following integral $$ \int_C ...
2
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1answer
28 views

to find the extreme values of function..

can anyone just help me with the below stated problem: Show that: $1.)$ $\text{sin}(x)+\text{sin}(y)+\text{sin}(x+y)$, $x,y\in [0,\pi/2]$ has a global maximum $3\sqrt3/2$ at $(\pi/3,\pi/3)$ and ...
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2answers
12 views

Bounded - Continuous Relation

How to solve the following question? $$$$ Suppose $f:A\subset\Bbb{R}^2\to\Bbb{R}$ continuous in the rectangle $A=\{(x,y)\in\Bbb{R}^2|\alpha\leq x\leq\beta;\alpha'\leq y\leq\beta'\}.$ Proof that $f$ ...
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1answer
23 views

$f(x,y)=2x+4y-x^2y^4$ has a critical point but no local extreme points.

I've to show that: $f(x,y)=2x+4y-x^2y^4$ has a critical point but no local extreme points. we mean by a critical point as that interior point where $f_x=f_y=0$ or the points where $f_x$ or ...
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0answers
23 views

Using polar co-ordinaries on an integral whose domain is a disk not centred at the origin

I need to find the volume of the region in $z > 0$ that lies within the cylinder $x^2 + y^2 = 2x$ and is bounded by the cone $z^2 = x^2 + y^2$. I have been struggling to set up the integral for ...
3
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3answers
106 views

Fundamental limit in two variables

Can I write that $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{u\to0}\frac{\sin(u)}{u}$$ and, hence, that $\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=1$? If so, why can I do it?
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0answers
32 views

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
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votes
2answers
51 views

How to parametrise the curve [closed]

C : curve of intersection of sphere centered at (1,1,0) and radius sqrt2 and the plane X+Y=2 direction of curve is taken as such that it begins at (2,0,0) goes below the XY plane and then comes to ...
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2answers
18 views

Doubt in proving differentiable when both partial derivatives are equal

I had a problem with a step in this: I have to prove that: $|xy|^{\alpha}$ is differentiable at $(0,0)$ if $\alpha > \frac{1}{2}$. In this case both partial derivatives exist and have the ...
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1answer
30 views

what is divergence? [closed]

I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To give you a sense of ...
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0answers
14 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
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1answer
21 views

calculating partial derivatives at $(0,0)$

Let $f:\mathbb R^2 \to \mathbb R$ given by := $$f(x,y) = \begin{cases} 0 & \text{, if xy=0 } \\ 1 & \text{, if xy $\neq$ 0} \end{cases}$$ I've to show that $\partial_1 ...
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2answers
29 views

If a continuous function of two variables has finite zero set, then it does not change sign

If $f$ is a continuous function from $\mathbb R^2 \rightarrow \mathbb R$ such that $f(x)=0$ for only finitely many values of $x\in\mathbb R^2$. Can we conclude that $f(x)\leq 0$ or $f(x)\geq 0$ for ...
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0answers
27 views

How to parametrize the volume of the intersection of cube and a right tetrahedron?

This is an extension of my previous question. I am trying to find the volume of the region which is the intersection of a cube given by $\vec r_1 = (x,y,z)$, where $$\begin{cases}0 \le x \le 1 \\ 0 ...
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2answers
24 views

The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
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1answer
39 views

Why can I not combine integrals this way?

Evaluating the triple integral $\int^1_0 \int^{1-z}_0 \int^{1-y-z}_0 \text{dxdydz}$, I get $\frac 16$. Evaluating the triple integral $\int^1_0 \int^1_0 \int^1_0 \text{dxdydz}$, I get $1$. So I ...
0
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2answers
33 views

Proving Multivairble Limit Exists [duplicate]

How do you deal with multivariable limits? We'll use the example $f: \mathbb R ^2 \rightarrow \mathbb R$ $$\lim _{(x,y) \rightarrow (0,0)}\frac{\sqrt{|xy|}}{\sqrt{x^2 + y^2}}$$ The limit doesn't ...
2
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1answer
31 views

Existence of partial derivative

I know how to compute partial derivatives of functions with more than one variable. But how can i assert that the partial derivatives of a given function exist at a point without computing it? ...
0
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1answer
38 views

I need help finding the mean radius of a cylinder

The question is: What is the mean radius $\overline{r}$ from the midpoint of a cylinder of radius $a$ and height $h$ to its boundary surface? Evaluate $\overline{r}$ for $a = h/2 = 10~\mathrm{cm}$. ...
0
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1answer
31 views

Is this vector identity accurate?

Does this identity hold true for vectors $A$, $B$ and the gradient operator? $(\nabla \cdot A)B = (A\cdot \nabla)B + (B\cdot \nabla)A$
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1answer
11 views

Limit equivalence

"Let $f:A\subset\Bbb{R}^n\to\Bbb{R}$ be a function and denote $\Bbb{x}=(x_1,\dots,x_n)$ and $\Bbb{p}=(p_1,\dots,p_n)$. Show the following equivalence: ...
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2answers
19 views

(Inequality) $p \cdot (z-x) \leq \frac{a}{R} | z- x| \Leftrightarrow |p|\leq \frac{a}{R}$

I need to solve this inequality: Let $z \in \mathbb{R}^N$ and $a,R > 0$, prove that $$(\forall x\in B_R(z)) \quad p \cdot (z-x) \leq \frac{a}{R} | z- x| \ \Longleftrightarrow \ |p|\leq ...
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2answers
34 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
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1answer
22 views

Doubt on understanding continuity .

Just preparing for my multivariable-calculus exam and wanted to clear these things: I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the ...
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0answers
11 views

Unit tangent vector

Let $f:I\to \mathbb R^3$ a vector valued function. When we define the unit tangent vector: $T(t)=f´(t)/||f´(t)||$ , $||f´(t)||\neq 0$ is it neccesary that $f$ is a $C^1$ function? or just ...
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0answers
11 views

Regarding functions from R² to R: continuity and differentiability

Let $f : U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^2$ is an open set and $P \in U$. I am almost sure the following statements are correct, but please confirm: The only requirement for ...
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2answers
28 views

Showing a function is convex on $x^2+y^2\leq a^2$

This is a question from my assignment about which I have no idea: Let $f(x,y)=\phi(x^2+y^2)$,where $\phi$ is of class $C^2$ ,increasing and concave. Show that $f$ is convex on disk $x^2+y^2\leq a^2$ ...
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1answer
21 views

Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$

Can anyone help with this: Let $x\in \mathbb R^n$ and $u=f(r)$,where $r=\|x\|$ and f is differentiable . Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$ . I can't ...
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2answers
148 views

Finding the limits of a multivariable function

Given the following function, determine whether the following function is continuous at $(0,0)$ $$f(x,y)=\begin{cases}\frac{x^2y^2}{x^4+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ ...
2
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0answers
39 views

Normalize gradient

I want to minimize a function $f \, : \, \mathbb{R}^{N} \, \longrightarrow \, \mathbb{R}$ (with $N \in \mathbb{N}^{\ast}$. In my problem, $N = 315$). I know that $f$ is differentiable on ...
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0answers
15 views

Question involving Stokes' theorem

Given a cector field F= (x^2-y^2 , -x^2 + y^2 , z ) S: portion of surface x^2+y^2 -2by + bz =0 whose boundry lies in xy plane . here im to evaluate doule integral of curlF.n dsigma ....from stokes ...
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0answers
31 views

A PDE problem related to the ratio of populations

Let $m>0$ in $\bar{\Omega}$ be a given nonconstant function, where $\Omega\subset \mathbb{R}^n$ is a bounded smooth domain. Then consider the following elliptic modeling problem: $$ \Delta ...
0
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1answer
32 views

Why i got negative value for volume?

I want to find the indicated volumes under the surface $z=\frac{1}{y+2}$ and over the area bonded by $y=x$ and $y^2+x=2$. After sketching the graph for $x=2-y^2$, and $x=y$ i found that $y=0$ and ...
0
votes
1answer
28 views

Find the point on a parameterized line closest to another line

Let $x_1 = (1, 2, 3)$ and $x_2 = (3, 2, 1)$. Consider the two lines $x_1(s) = x_1 + su_1$ and $x_2(t) = x_2 + tu_2$. $u_1 = (\frac{2}{\sqrt{5}}, \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}})$, $u_2 = ...