Tagged Questions

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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0
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0answers
7 views

A question from calculus my test(Curl, guess theorem )

the value of the integral $$ \iint rotF*n*ds \quad where \quad s-> x^2+y^2+z^2=4 \quad $$ and the normal is making a blunt angle with the Z axis, and $$ f=(zsinx-2y+1)i+(3x)j+(4xz+z^3)k $$ im ...
0
votes
1answer
28 views

Splitting partial derivatives

How come $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}$$ when $$ u = x\; cos \theta - ...
0
votes
2answers
29 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
4
votes
2answers
169 views

Can a region always be parametrized by a single function?

Can some connected region in $\Bbb R^n$, possibly with some other nice conditions on the region, always be parametrized by a single function $\vec r(x_1, x_2, \dots, x_k)$ (even if it may be easier to ...
0
votes
1answer
17 views

Gradient of an implicitely defined function?

For some function $F(x,y,z) = 0$, is the gradient $\nabla F $ always equal to zero? If you take the partial derivatives of both sides, you get zero for all of them. My book says: Which implies that ...
0
votes
0answers
9 views

Lagrange Multipliers

The Question: Find the minimum distance between the origin and the surface $x^2y -z^2 +9 = 0$. I've been able to find the critical points when $x =0$ and when x is not equal to zero but lamda is ...
0
votes
1answer
22 views

Is this function of 2 variables differentiable?

$f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2}$ when $(x,y) \neq (0,0)$ and $0$ when $(x,y) = (0,0)$ Is f differentiable?
1
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2answers
21 views

Chain rule for multiple variables?

What I've tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial ...
1
vote
2answers
36 views

Contradiction when differentiating?

Consider the function $F = x+y$. Let $x = t$ and $y= \cos t$. By directly differentiating, $$\frac{\partial F}{\partial x} = 1$$ and $$\frac{\partial F}{\partial y} = 1$$ Using the chain rule ...
0
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0answers
26 views

Question about a proof in Lang's undergraduate analysis

This is from page 580 of Lang's undergraduate analysis (2nd edition). I have difficulty in understanding the proof, hope that someone here can enlighten me. My questions are: i) On line 5 of the ...
0
votes
1answer
22 views

Geometric interpretation of derivative?

For some function $F(x,y) = 0$, $$\frac{dy}{dx} = \frac{-F_x}{F_y}$$ Can someone give me a geometric interpretatio of this? ($F_x$ and $F_y$ are the partial derivatives)
0
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0answers
20 views

Lagrange multiplier, how to show that these two methods gives the same solutions.

I have read about another way of using Lagrange multipliers, but I can not explain why this is the same as I have seen before. I have seen this before: Lets say you want to maximize ...
2
votes
2answers
24 views

How to check extrema if second derivative test fails

I have to find minima and maxima of $f(x,y)=x^4+6y^2-4xy^3-1$ I found three points that could be extrema - $(0,0)$, $(1,1)$ and $(-1,-1)$ I already checked $(1,1)$ and $(-1,-1)$ with second ...
1
vote
0answers
60 views

Finding the volume between a paraboloid and plane.

I have to find the volume between the plane $z=3-2y$ and the paraboloid $z=x^2+y^2$. I understand that the domain of this region is the disk centred at (0,-1) with radius 2 if I set the equations ...
0
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0answers
20 views

Pullback of Euclidean metric under spherical coordinates

I came across the following problem when I was reading a paper. Let $\Omega=B_2\subset\mathbb{R}^3$, which is the ball of radius 2 centered at origin. Map $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
1
vote
2answers
50 views

How to show that the curve $ (x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $ is an ellipse?

Show that the curve $$(x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $$ is an ellipse in the plane it lies on. $$x^2 + y^2 = (\sin t)^2 + (\cos t)^2 = 1$$ $$x^2 + (z/c)^2 = (\sin t)^2 + (\cos ...
0
votes
2answers
19 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...
0
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0answers
7 views

mativariate analysis and clusters

the last question , solve by hand and by R programming
0
votes
0answers
10 views

Euclidean distance matrix , ingle linkage and compelete linkage cluster.

consider the following 5*2 data matrix, -1 1 A= -1 -1 0 0 1.3 0 2.3 0 working by hand and R- programming: a) calculate the Euclidean distance matrix between the 5 ...
1
vote
1answer
11 views

Does continuity of composition of maps gives continuity of the left function?

Suppose $\;F:\Bbb R^2\to\Bbb R\;$ is such that for any continuous path $\;\gamma:[0,1]\to\Bbb R^2\;$ , the composition $\;F\circ \gamma:[0,1]\to\Bbb R\;$ is continuous . Is then $\;F\;$ continuous? ...
2
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2answers
32 views

How to prove this multivariable integral identity?

By numerical experimentation I found that $$ \lim_{\beta \rightarrow \infty} \frac 1 \beta \int_0^{\beta}dx \int_0^{\beta}dy \, f\left( |x-y| \right) = 2\int_0^{\infty}dx \, f(x) $$ if $f:\mathbb{R} ...
0
votes
0answers
16 views

Confusion in this line integral

Given vector field (M,N,P) with components as M=3y^2 + 2z^2 , N= (6x - 10z)y , P=4xz -5y^2 along portion from (1,0,1) to (3,4,5) of Cis intersection of two surfaces z^=x^2 + y^2 (cone) and z=y+! ...
0
votes
0answers
15 views

Changing a double integral to single integral

I have seen this integration problem in a random process text book. We have the following integral. $\int_{-T}^{T}\int_{-T}^{T}C(t_1-t_2)dt_1dt_2 = \int_{-2T}^{2T}(2T-|\tau|)C(\tau)d\tau$ where ...
0
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0answers
8 views

How many distinct partials of order $k$ for a function $f: \mathbb{R}^{n}\rightarrow\mathbb{R}$?

Studying for the math subject GRE, and I come across the titular question. I didn't take any combinatorics or probability courses in college, and I'm realizing I have no intuition for counting. Could ...
0
votes
1answer
21 views

Find $f,g$ for a counterexample of multivariable limit

Are there any $f,g : \mathbb R^2 \rightarrow \mathbb R$ such that $\lim_{x \rightarrow 0} f(x) = 0, \lim_{y \rightarrow 0}g(y) = 0$ but $$ \lim_{(x,y) \rightarrow (0,0)} \dfrac{\log(1+f(x)g(y))}{g(y)} ...
0
votes
1answer
27 views

Proof of $\iint \limits_{\delta V} f \overrightarrow{dA} = \iiint \limits_V \nabla f \, dV$

In my aerodynamics class, we often use the identity: $\iint \limits_{\delta V} f \overrightarrow{dA} = \iiint \limits_V \nabla f \, dV$ for a closed surface (can't seem to get \oiint to work) and ...
0
votes
0answers
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
votes
1answer
34 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 ...
1
vote
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/
1
vote
1answer
25 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
votes
2answers
28 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
vote
1answer
25 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 ...
0
votes
1answer
28 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 $$xe^{u(x)}+u(x)e^{v(x)}=0=xe^{v(x)}+v(x)e^{u(x)}$$ ...
0
votes
1answer
25 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 ...
0
votes
0answers
16 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 ...
0
votes
1answer
17 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 $(0, 0, 6)$: $$x + y + z = 6e^{xyz}.$$
1
vote
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
votes
1answer
31 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 ...
-1
votes
0answers
17 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 $M>0$ ...
1
vote
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 ...
0
votes
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 ...
0
votes
3answers
42 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
vote
1answer
29 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 ...
3
votes
1answer
93 views
+150

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 ...
1
vote
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 $f$ ...
0
votes
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 ...
0
votes
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
votes
3answers
112 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?
3
votes
1answer
70 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 ...
-3
votes
2answers
63 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 ...