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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3
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3answers
323 views

Prove an analog of Rolle's theorem for several variables

On p. 135 of Buck's Advanced Calculus, he asks the reader to prove an analog of Rolle's theorem for functions of two variables (I suspect the number two is arbitrary). The hint is to assume that $f=0$ ...
2
votes
1answer
104 views

Integral on surface

Please help me calculate integral on surface. $$\iint\!(2-y)\,dS $$ $$ 0\leq z\leq 1;\, y =1 $$ I can't understand what should I do with '$z$' coordinate. I assume I should do something with it, not ...
2
votes
2answers
33 views

What is the definition of curl of $\mathbf{F}(x_1, x_2) = ( F_1 (x_1,x_2) F_2(x_1,x_2))$ in $\mathbb{R}^2?$

What is the definition of curl of $\mathbf{F}(x_1, x_2) = ( F_1 (x_1,x_2) F_2(x_1,x_2))$ in $\mathbb{R}^2?$ Most textbook says only of vector fields in the space $\mathbb{R}^3$...
4
votes
2answers
347 views

Calculating area of astroid $x^{2/3}+y^{2/3}=a^{2/3}$ for $a>0$ using Green's theorem

question as follows. Show that for any planar region $\Omega$, $$\mathrm{area}\left(\Omega\right)=\frac{1}{2}\oint_{\partial\Omega}(xdy-ydx).$$ Use this result to find the area enclosed by the ...
0
votes
2answers
65 views

Bounds of $\oint_{\partial R}\left((x^2-2xy)dx+(x^2y+3)dy\right)=\iint_{R}\left(2xy+2x\right)dxdy$

I'm just having some trouble figuring out the bounds and boundary of the following integral. Question as follows: Evaluate $$\oint(x^2-2xy)dx+(x^2y+3)dy$$ around the boundary of the region contained ...
3
votes
1answer
2k views

Distance of a test point from the center of an ellipsoid

I'm trying to learn about Mahanalobis distance and I'm pretty close to getting the idea. I've learned that the distance has got a lot to do with the properties of an ellipsoid. I have understood so ...
0
votes
1answer
196 views

equation of the tangent plane

Find an equation of the tangent plane to the surface at the given point. $g(x, y) = x^2-y^2$ at $(7, 2, 45)$. I know the answer is between $14(x-7)-4(y-2)+(z-45)=0$ or $14(x-7)-4(y-2)-(z-45)=0$. I ...
2
votes
2answers
131 views

extrema and saddle points

Examine the following function for relative extrema and saddle points: $$f(x, y) = 9x^2-5y^2-54x-40y+4.$$ I did this and got that the point should be at $(3, -4, 3)$. Is that right? Also, how do I ...
1
vote
3answers
58 views

Gradient of a function in multi-variate calculus please help

Find the gradient of the function at the given point. $g(x, y) = 3xe^{y/x}$, at point $(3, 0)$. How do you compute the gradient of this function. Please help me.
0
votes
1answer
31 views

Computing directional derivative of $f(x,y,z)$

Find the direction derivative of $f(x,y,z) = xy + yz + xz$ at the point $P(1,1,1)$ in the direction of $v = \langle7,3,-6\rangle$. I got the answer as $32/\sqrt{94}$. Is that right? If not what is the ...
1
vote
1answer
245 views

Find the equation of the tangent plane given a vector instead of point

Find the equation of the tangent plane at $\mathbf p = (0,0)$ on the surface $z=f(x,y)=\sqrt{1-x^2-y^2}$. Give an intuitive geometric argument to support the result. However $\mathbf p$ is a ...
1
vote
1answer
644 views

How to examine if multivariable functions are differentiable?

How to examine if functions: $f(x,y)=|x+y|$ and $g(x,y)=\sqrt{|xy|}$ are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$
0
votes
2answers
114 views

How to check if function has global extremes?

I need to determine if function: $f(x,y)=x+2y-2\log(xy) $ has global minimum/maximum. I've found local minimum at $(2,1)$, but that's not any proof of global minimum.
1
vote
1answer
84 views

I'm looking for a function $f(x,y,z)$, which has partial derivatives only in single point

Function must be defined in $\mathbb R$. I know that Dirichlet function is involved somehow, but i still can't find out an example.
3
votes
3answers
154 views

Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...
6
votes
1answer
226 views

Show $\nabla\cdot\left(\mathbf{F}\times\mathbf{G}\right)=\mathbf{G}\cdot(\nabla\times\mathbf{F})-\mathbf{F}\cdot(\nabla\times\mathbf{G})$

Question as follows. Suppose that $\mathbf{F}$,$\mathbf{G}:\mathbb{R^3}\rightarrow\mathbb{R^3}$ and $\phi:\mathbb{R^3}\rightarrow\mathbb{R}$ are smooth. Show using the summation convention that ...
3
votes
2answers
87 views

Chain rule for functions of two variables

Suppose that $f(x,y)$ is a function of two variables with $f_x(0,2) = 2$ and $f_y(0,2) = -1$. Using the chain rule compute the numerical value of $f_\theta(r\cos\theta,r\sin\theta) = 2$ at $r=2$, ...
3
votes
2answers
384 views

Lipschitz continuity and sup of derivative norm

On this wikipedia page, it is stated: For a differentiable Lipschitz map $f : U \rightarrow R^m$ the inequality $\|Df\|_{\infty,U}\le K$ holds for the best Lipschitz constant of $f$, and it ...
6
votes
2answers
173 views

Proving that $\iint_S (\nabla \times F) \cdot \hat{n} dS =0$

I have the following question: Prove that $$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =0$$ for any closed surface $S$ and twice differentiable vector field $\vec F:\mathbb{R^3} \to ...
2
votes
1answer
46 views

How The Jacobian of the transformation can be shown to not depend on $X_i$ or $\bar X $ and is equal to the constant $n$

Transform the random variables, $X_i$, $i=1,2,\ldots,n$ to $$ \begin{align} Y_1 & =\bar X \\ Y_2 & =X_2-\bar X \\ Y_3 & = X_3-\bar X \\ & {}\ \vdots \\ Y_n & =X_n-\bar X ...
6
votes
1answer
57 views

how to prove a parametric relation to be a function

For example lets suppose that I have given the functions $f:\mathbb{R}\longrightarrow \mathbb{R}$ and $g:\mathbb{R}\longrightarrow \mathbb{R}$. If my relation is $R=\{(x,(y,z))\in \mathbb{R}\times ...
8
votes
1answer
113 views

Surface integral of $2x+y+2z=16$

Here's the question: Find the surface area of the part of the plane $2x+y+2z=16$ bounded by the surfaces $x=0$, $y=0$ and $x^2+y^2=64$. So, I know I have to parameterize the surface ...
3
votes
1answer
383 views

Positive homogeneous

Question: A set $X\subset\mathbb{R}^n$ is called a cone in $\mathbb{R}^n$ if $x \in X$ implies $tx ∈ X$ for every $t\in\mathbb{R}_+$. Given a cone $X$ in $\mathbb{R}^n$, a function $f\colon X ...
1
vote
0answers
43 views

Which of these statements about multivariable limits are true?

Let $f :\mathbb{R}^2 \to \mathbb{R}$ be a map. Then a) $\lim\limits_{x\to 0} \lim\limits_{y\to 0} f (x, y)$ exists implies $\lim\limits_{(x,y)\to 0} f (x, y)$ exists. b) ...
1
vote
2answers
219 views

Parametrization of $x^2+y^2=z^2$

How can we show that any point on $x^2+y^2=z^2$ can be written in the form (z $\cos(\theta)$, z $\sin(\theta)$, z) for some $\theta$? Here is how I tried to approach it: $$(z \cos(\theta))^2+(z ...
3
votes
3answers
129 views

How do you formally prove that a function in several variable is really a function

Let say for example that we define $f:\mathbb{R}^{3}\longrightarrow \mathbb{R}^{3}$ such that $f(x,y,z)=(y^{2},xz,xy^{2})$. My informal argument would be just that there is only one object that can ...
3
votes
1answer
78 views

Simplifing formulas using tensor notation

Im trying to symplify formulas like: $$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$ or something more strange like: ...
1
vote
1answer
147 views

Differentiability of multivariable functions, and its relation to the chain rule.

I'm struggling with the conditions for the applicability of the chain rule. $${df(C(t))\over dt} = \mathrm{grad}f(C(t))\cdot C'(t)$$ Where $C$ is in $\Bbb R^n$ and $f$ is differentiable in the ...
3
votes
0answers
53 views

Uniform estimate for multivariate Taylor's formula

I am wondering how the remainder in multivariate Taylor's formula could be uniformly bounded in a small ball around a point. For instance, let $f:\mathbb R^n\rightarrow\mathbb R^m$ be a function of ...
3
votes
1answer
64 views

Find the image of the set

Given set $$D=\{(x,y): x\geq -1, y-x\geq1, y+x\leq1\}$$ and the regular transformation $$\phi:\begin{cases}u=x^2+y^2\\v=x+y\end{cases}$$ How to find out the image of $D$, i.e. $\phi (D)$?
2
votes
1answer
65 views

Reference for an integral's convergence on an $n$-ball when $n>2$.

I was searching for a reference of a standard result from calculus. Unfortunately I couldn't find it. I think that's mostly due to I am not familiar with any english calculus book. So I am ...
3
votes
2answers
537 views

Integrating velocity field to get position

I feel silly for simply being brainstuck, but consider the following integral, physically it would be the solution of $\mathbf{p} = \tfrac{d\mathbf{v}}{dt}$ - the position of a given particle in ...
4
votes
2answers
205 views

Prove an identity about $\iint_S\mathbf{r}\wedge d\mathbf{S}$ using Stokes' theorem

$$ \int_C\mathbf{r}(\mathbf{r}\cdot d\mathbf{r})=\iint_S\mathbf{r}\wedge d\mathbf{S} $$ With $\mathbf{r} = (x,y,z)$ being a 3-dimensional vector. How do you get this result using Stokes' theorem?
1
vote
4answers
57 views

dot products in a linear algebra context

I've been self-teaching myself linear algebra from Linear Algebra and its Applications 4th from D. Lay. I'm about 8 sections deep and I've had this bothersome feeling regarding the section describing ...
0
votes
2answers
82 views

Statistics problem (normal distribution)

A factory produces roller stands (of cylindrical form) that has $4cm$. of diameter and $6cm$. of length. In fact, the diameters $X$ are normally distributed with a mean of $4cm$. and a standard ...
2
votes
1answer
142 views

How would one work this out (surface integral of a function)

Given the definition of the surface integral: $$Area=\iint_{S}{\mathbf{F}\cdot d\mathbf{S}} = \iint_{D}{f(\mathbf{r}(u,v))\cdot \left |\mathbf{r_u} \times \mathbf{r_v} \right |dA}$$ Where ...
1
vote
1answer
167 views

Proving minima, maxima, and saddle points don't exist.

I've got the next function: $f(x,y)=x^3y^3$ where $x,y\in \mathbb R$ I need to determine whether there is minima, maxima or saddle point. Easily enough, after doing the partial derivatives ...
1
vote
1answer
698 views

True Velocity and Heading

An airplane flies at $670$ MPH directly northwest. Wind blows at $70$ MPH from the west (i.e the wind is blowing towards the east). Determine the true velocity and heading of the plane. Steps: ...
1
vote
2answers
46 views

Doubt with bounds of the following integral:

Hi question is as follows: use polar coordinates to evaluate the following integral $$\int_{0}^{2}\int_{-\sqrt{2y-y^2}}^{\sqrt{2y-y^2}} \sqrt{x^2+y^2} dxdy.$$ I'm stuck finding the limits for $r$. ...
1
vote
2answers
67 views

Simple partial differentiation

I have a simple partial differentiation question here, given: $u = x^2 - y^2$ and $v = x^2 -y$, find $x_u$ and $x_v$ in terms of $x$ and $y$. What is the easiest way to go about this? Thanks
0
votes
1answer
89 views

global extrema for 2 variable multi var calc

Find all global extrema of f(x, y) = x^2 + y^2 on the region s = {(x,y) : x^2 + y^2 <= 1} I did this would this be a global minimum at (0,0) what do you have to do when it says on the region.
0
votes
2answers
276 views

Converting a point to Cylindrical and Spherical Coordinates

How is any point on the Cartesian coordinates converted to cylindrical and spherical coordinates. Taking as an example, how would you convert the point (1,1,1)? Thanks in advance.
1
vote
1answer
61 views

How to find all local extrema using multi-variate calculus

Find all local extrema of $f(x, y) = x^2 + y^2$. That is, find their locations and values. I started this problem but when I took the first derivative I got the critical points as $(0, 0)$. Then ...
1
vote
1answer
120 views

Surface integrals of second kind

In the formula for calculating surface integrals of second kind, we have: But, this integral is denoted by $\int \int _S \vec{F}\cdot \hat{n}dS $ . So, should we always normalize the expression $ ...
2
votes
1answer
64 views

For given $p$, a regular mapping $F:M \to N$ of surfaces can be diffeomorphism

Want to show : For given point $p$ of M, a regular mapping $F:M \to N$ of surfaces has a neighborhood $U$ such that $F|_{U}$ is a diffeomorphism of $U$ onto a neighborhood of $F(p)$ in N. I learned ...
3
votes
0answers
72 views

Tangent bundle of a surface is a manifold

My differential geometry textbook defined the tangent bundle of a surface as the set of all tangent vectors to M at all points of M. The abstract patches are also given : ...
1
vote
1answer
96 views

A question concerning partial derivatives

Let $F(x-y,y-z,z-x)=0$,find $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$. This is a homework problem,I don't know how to do,appreciate any help.
1
vote
2answers
47 views

Area of a parallelogram using Cross Product

How do you compute the area of the parallelogram with 4 arbitrary corners, say at (1,1,1), (2,3,4), (3,2,1), and (4,4,4) using a cross product? I understand with 3 corners but getting a little lost ...
0
votes
5answers
60 views

what is this value called?

I've come across this operation that takes in two vectors $a=(X_{a},Y_{a})$ and $b=(X_{b},Y_{b})$ and returns a single number $X_{a}\cdot Y_{b}-Y_{a}\cdot X_{b}$ Is this like 'a thing'? Like the ...
0
votes
1answer
36 views

Check for directional derivative and show that $f(1,1,1) > f(0,0,0)$ when given the partial derivatives

I'm kinda lost in this exercise Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be of class $C^1$ and $\forall x=(x_1,x_2,x_3)\in\mathbb R^{3}$ $$\frac{\partial f}{\partial x_{1}}(x)=x_{2}, ...