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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26 views

What is the variance E[A]^2, statistics? [closed]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
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1answer
20 views

Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
2
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1answer
46 views

Inverting a function

I am stuck with the following problem I am supposed to find the inverse of the function $g$ with $2$ variables, where $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
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1answer
37 views

How to find the normal vector of $xyz=1$

How do I find the normal vector of $ xyz=1 $ at $(a, b, c)$? Is the answer below correct? Because some answers on here are saying that the normal vector is $$ \Delta F = (f_x,\:f_y,\:-1) $$ So ...
2
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0answers
29 views

Directional derivatives with given values.

At the point (1,2), the function f(x,y) has a derivative of 2 in the direction toward (2,2) and a derivative of -2 in the direction toward (1,1). Find f_x(1,2) and f_y(1,2). Find the derivative of f ...
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0answers
12 views

Divergence using suffix notation

I am trying to show the divergence of $v(r) = ∇r^n$ where $r =$ ||r|| using suffix notation. The solution I am given says: $∇ · v(r) = ∂^2_ir^n = n∂_i(r^{(n−2)}r^i) = n((n − 2)r^{n−4}r^2+i + ...
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0answers
9 views

general inverse of multivariable function

Given $m<n$. Let $f:N\subset\mathbb R^m\mapsto \mathbb R^n$ be a differentiable function. I am looking for the condition(s) such that we can find a function $g:im(f(N))\mapsto \mathbb R^m$ ...
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1answer
18 views

Determining when $f(x,y) = x^{4/3} \sin(y/x)$ ($x \ne 0$) is differentiable.

Let $f$ be defined as follows: $$ f(x,y) = \left\{ \begin{array}{lr} x^{4/3} \sin(y/x) & \mathrm{if\ } x \ne 0 \\ 0 & \mathrm{if\ } x = 0. \end{array} \right. $$ I am asked to determine where ...
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1answer
16 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
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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 ...
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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 - ...
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1answer
16 views

Finding extrema of functions with 2 variables. [closed]

Find the largest sphere inscribed inside tetrahedron witn vertices (1,0,0), (0,1/2,0), (0,0,1), and (0,0,0). Round the radius to 2 decimals.
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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, $f'>0$. ...
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2answers
166 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 ...
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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 ...
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0answers
8 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 ...
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1answer
21 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?
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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 ...
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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 ...
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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 ...
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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)
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19 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 ...
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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 ...
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0answers
57 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 ...
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0answers
18 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 $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
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1answer
68 views

Triple integration and transformations [closed]

Hello I am a 17 year old kid in high school that does math for fun. One of my buddies gave me these problems and I can't seem to figure it out. Question 1 Find the volume of the finite region in ...
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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 ...
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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 ...
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7 views

mativariate analysis and clusters

the last question , solve by hand and by R programming
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9 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 ...
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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? ...
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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} ...
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15 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+! ...
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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 ...
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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 ...
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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)} ...
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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 ...
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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 ...
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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
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 $DF=DfDg$ ...
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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 ...
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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 ...
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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
23 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
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 $(0, 0, 6)$: $$x + y + z = 6e^{xyz}.$$
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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
30 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
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 $M>0$ ...