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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1answer
80 views

Fréchet differentiability of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ I want to determine ...
2
votes
0answers
36 views

Find $F'(x)$ where $F(t) := \int_{g(x)}^0 f(x,t)dt$

Let $f: \mathbb{R^2}\to \mathbb{R}$ be $C^1$ and $g:\mathbb{R}\to \mathbb{R}$ also $C^1$. Define $$F(t) := \int_{g(x)}^0 f(x,t)dt$$ Find $F'(x)$. What I did was first define $\Phi : \mathbb{R}\...
0
votes
2answers
32 views

vector and curl identity

This popped up in my notes and the author made no remarks about the properties used $\bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0}$ Then, $\...
0
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0answers
18 views

higher derivatives of $R^m \to R^n$ [duplicate]

What's a good source (paper, book, website,...) where I can learn more about higher derivatives of functions $R^m \to R^n$ as multilinear functions or tensors? Thanks.
0
votes
1answer
19 views

What is $(A_1 \times … \times A_n) \cup(B_1 \times … \times B_n)=?$ ,$A_i$'s are intervals

What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively. What I mean is can $(A_1 \times ... \...
1
vote
2answers
49 views

What is the measure of $A=[-1,2]\times[0,3]\times[-2,4]\cup[0,2]\times[1,4]\times[-1,4] \setminus [-1,1]^3$?

I really get stuck after one point, and don't know where to go on.I know that my try, up to where I am stuck is correct. $$\color{#20f}{\text{TRY:}}$$ $$B_1=[-1,2]\times[0,3]\times[-2,4],\mu(B_1)=3 \...
0
votes
1answer
34 views

Triply integral involving spherical coordinates - how can I proceed?

$$ \iiint_V \frac{1}{x^2+y^2 + z^2 } dx dy dz =? $$ where $$ V=\{ (x,y,z)| x^2 + y^2 + (z-1)^2 \leq 1 \}. $$ After moving to spherical coordinates I obtain: $$ \iiint \sin \theta dr d\theta d\phi $$ ...
2
votes
1answer
40 views

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$.

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$. The volume I'm asked to find it's what is left of the unit semisphere minus the upper ...
0
votes
2answers
29 views

Slope of the tangent of the curve $cos(x-y)+sin(x+y)=1$, when $x=y=\pi$

Now I have graphed this online and get Now by directly substituting, I get $cos(0)+sin(2\pi)=1$ which is just $1=1$. So what exactly is this question asking?
3
votes
3answers
124 views

Continuity of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$? [duplicate]

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ Is this function ...
-1
votes
2answers
50 views

Find the tangent plane on $z=x^3-xy$ perpendicular to $(1,1,1)$

I'm not sure how to do this. I tried letting $\partial{z}/\partial{x}=1$ and $\partial{z}/\partial{y}=1$ then solving for $z$ at this point and subbing them into $x+y+z=c$ but I just get $x+y+z=0$
0
votes
2answers
52 views

How do I determine the maximum or the minimum of in the range the function in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$?

How do I determine the maximum or the minimum of in the range the function $f(x,y)=x^3+y^3 $in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$? As a continuous function f must ...
2
votes
1answer
40 views

Multidimensional taylor series $sin (x^3y^2) $

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
1
vote
1answer
48 views

How to do a Taylor expansion of a vector-valued function

Let $f:\Bbb R^2\to \Bbb R^2$ be given by $$f(x,y):= \left(e^x\sin(x+y),e^{y-x}\tanh(y)\right)$$ Find the second-order Taylor expansion of $f$ about (x,y)=(0,0)$. I know how to find the Taylor ...
0
votes
2answers
34 views

How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $ I need to make Jacobian matrix of $f\circ g$. I found ...
-1
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0answers
31 views

What if curl of vector field has more than 1 dimension?

Assume $v(x,y,z) = (xy, -2yz, z^2 - yz)$, then this field's curl is $(2y-z, 0, -x)$. So what am I to make of this, concerning the plane where the rotation occurs? It is known that if a field's curl ...
1
vote
1answer
33 views

Help needed understanding this explanation for the Jacobian

In this Quora Answer, a very intuitive explanation for the Jacobian is provided, there is however a step I don't understand: He takes this square: And via a polar coordinates transformation ...
0
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0answers
25 views

Instantaneous rate of change of a three-dimensional parametric

In physics, it is common to define the horizontal position $x$ and the vertical position $y$ of an object as functions of $t$ and then us the formula $$\dfrac {dy}{dx}=\dfrac {\dfrac {dy}{dt}}{\dfrac {...
1
vote
1answer
54 views

Evaluate double integral $\int_{}^{} \int x^2 cos(x^2-xy)dxdy$

Evaluate double integral $\int_{}^{} \int x^2 cos(x^2-xy)dxdy$ where region is bounded by sides of triangle whose vertices are $(0.0)(1,0)(0,1)$ I used order $dydx$ to evaluate it. I becomes ...
3
votes
0answers
50 views

What is the index of a vector field with positive divergence?

Let $v:\Bbb R^n\to\Bbb R^n$ be a smooth vector field with an isolated zero at $z\in\Bbb R^n$. Suppose that $(\operatorname{div}v)(z)>0$. Can we say anything about the index of $v$ at $z$? I know ...
0
votes
1answer
18 views

How do you use curves to solve multivariable limits?

This is a very beginner problem that I've had ever since this concept was introduced in our class. Basically one of the methods to solve limits that was presented to us involved setting a variable ...
2
votes
2answers
16 views

What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
0
votes
1answer
22 views

Volume between planes and a cylinder

Exercise: Calculate the volume between $ x\geq 0, y \geq 0, z\geq 0, y^2 + z^2 =1, x=2y$. I know it's a routine exercise but I fail to draw a proper graph, so my result may be wrong. Can someone ...
3
votes
3answers
81 views

Evaluation of $\iint_D \frac {\ln(2 - \sin \xi \cos \eta)\sin \xi} {2 - 2\sin \xi \cos \eta + \sin^2 \xi \cos^2 \eta} \mathrm d\xi \; \mathrm d\eta$

Evaluate the following integral: $$\iint_D \frac {\ln(2 - \sin \xi \cos \eta)\sin \xi} {2 - 2\sin \xi \cos \eta + \sin^2 \xi \cos^2 \eta} \mathrm d\xi \; \mathrm d\eta$$ where $D = [ 0, \pi/2] \...
0
votes
0answers
26 views

What conditions are needed for this statement to be true and why?

In the summary of the lectures, appears the sentence: "Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, then $x_0$ is a local maximum of $f$ if and only if, for every $0\neq v \in \mathbb{R}...
1
vote
1answer
35 views

Find the flux integral

Find the flux integral of $$ F=(\cos xyz, \tan xyz, 1+\arctan xyz) $$ across the surface $x^2+y^2=1$, $0\leq z \leq2$ and oriented by normal vectors pointing away from the z-axis. I tried to do it ...
0
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0answers
15 views

Finding the rates of investment of every variable in a multi-variable equation to get the most efficient output.

Math is said to be about teaching problem-solving skills for all problems, but a lot of problems have too many factors to reliably get a result. So how exactly do you get the answer to an equation ...
0
votes
1answer
60 views

Computing Complex Line Integrals

I'm having trouble understanding exactly how to compute a complex line integral in $\mathbb{C}$. With my understanding of multivariable calculus, I view the line integral of a vector field $F: \mathbb{...
0
votes
1answer
52 views

When is the line integral independent of parameterization?

Let $\alpha: [a,b] \rightarrow \mathbb{R}^2$ be a smooth path (i.e. $\alpha'$ is continuous on $[a,b]$), and let $f$ be a continuous vector field. The line integral of $f$ along $\alpha$ is defined as ...
1
vote
0answers
48 views

Is this map an immersion?

Let $g:\mathbb{R}^2\to \mathbb{R}^4,\ (x,y)\mapsto ((2+3\cos(2\pi x))\cos(4\pi y),\ (2+3\cos(2\pi x))\sin (4\pi x),\ 3\sin(2\pi x)\cos(2\pi y),\ 3\sin(2\pi x)\sin(2\pi y))$ I have to prove that for ...
2
votes
4answers
62 views

Can we determine the injectivity of a map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ on subset of $\mathbb{R}^n$ by looking at the Jacobian?

Specifically, I'm looking for an analogue of the following theorem in the case of real functions: Let $f: A \subset \mathbb{R} \rightarrow \mathbb{R}$ be monotone on $A$, then $f$ is injective on $...
1
vote
1answer
30 views

Showing that a function $f:\Bbb R^2 \to \Bbb R$ is differentiable

How do we show formally, using the limit definition of the derivative, that the function $$f: \Bbb R^2\setminus\{(0,0)\} \to \Bbb R\; ,\;\; f (x,y) =\frac 1{x^2 + y^2} $$ is differentiable on its ...
0
votes
1answer
31 views

How to prove a property of a function using the monotonicty of its integrals?

Let $f:[0,1]^2 \rightarrow \{0,1\}$, $f_B(b) = \int_0^1 f(b, s)\; ds$ and $f_S(s) = \int_0^1 f(b, s)\; db$, such that $f_B$ is non-decreasing and $f_S$ is non-increasing. Define the function $\hat{f}...
0
votes
1answer
44 views

Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
3
votes
2answers
110 views

Computing $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$ in Spherical Coordinates

Compute: $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$. Hint given: show that $\cos \theta> {r^2+3\over 4r}$ $1<r<3$ What I already did: I shift the unit ...
1
vote
2answers
61 views

Calculating Gradient of a $2$ Variable Function

Let $f(x,y)$ have continuous partial derivatives at every point. We know that $$\nabla f(0,3)=5 {\bf{i}} - {\bf{j}}$$ Then we define $g(x,y)=f(x^2-y^2, 3x^2y)$. I am not sure what I should do to ...
1
vote
1answer
27 views

Supremum and infimum of function of two variables

Consider $D = \left \{ x \in \mathbb{R} : x_1^2 + 44x_2^2 \leqslant 5 \right \}$ and function $f: D \rightarrow \mathbb{R}$, $f(x) = 13x_1 - 22x_2$. Find supremum and infimum of $f$. For both of them ...
0
votes
2answers
36 views

Inverse image of a function in multivariable calculus?

Let $f: R^2 \rightarrow R^2 $ defined by $f(x,y) = (x+y,xy).$ Claim : Inverse image of each point in $R^2$ under f has at most two elements. My Claim : Suppose $f(x,y) = (x+y,xy)= (p,q).$ We have ...
0
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0answers
38 views

Find $\iint (\nabla \times F)\cdot dS$ if S the surface of the sphere $x^2+y^2+z^2=a^2$

Find $\iint (\nabla \times F)\cdot dS$ if $F= y i+(x-2xz)j- (xy) k$ and S the surface of the sphere $x^2+y^2+z^2=a^2 $ above of the $xy-$plane I do not know if I must use the stokes theorem or try ...
1
vote
3answers
69 views

Find the area between the cylinder $z^2+y^2=r^2$ and two planes

I'm having trouble with this problem: Find the surface area between the top of $z^2+y^2=r^2$ between $z=ax$ and $z=bx$ (consider $a \gt b \gt 0$). I think I must find the area between the ...
0
votes
0answers
24 views

Seeking help evaluating a definite Integral

I am trying to evaluate the following integral $$I = \int_1^t u^{2 \kappa -1} \exp\left(-\frac{1}{2} b \omega^2 u^{2 \kappa} + a \omega u \right) du.$$ Upon integrating by parts we have $$I = u ^{...
2
votes
1answer
36 views

Local minimums and maximums of function of three variables

I got such function: $f(x_1, x_2, x_3) = x_1 x_2 x_3(4-x_1-x_2-x_3)$ I need to find all local minimums and maximums of this function. I calculated partial derivatives and I got that the only points ...
1
vote
1answer
33 views

To solve large systems of multivariate polynomial equations

Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (...
-1
votes
0answers
29 views

Explaination of the proof of directional derivative formula

Hi guys :) i read that proof of the formula of the directional derivative and i didn't understand the sceond step and what is h(o) and where does it come from? Where does the dot product between the ...
0
votes
0answers
26 views

C alculating flux using the divergence theorem when the divergence is 0

I calculated the divergence of my vector field $\langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle$ to be $0$. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ...
0
votes
0answers
29 views

On inequalities related with $f(s):=-(1-\frac{2}{2^s})^{-1}$

My Question. a) How can you prove easily that the multivariable function in LHS is positive on $x^2+y^2<1$ $$2^{1-x}\cos(y\log 2)-1>0?$$ b) Let $s=\sigma+it$ the complex variable, ...
0
votes
0answers
22 views

integrating a function of a dirichelet random vector on a subset

Consider a random vector $\bar{X}$ of n many variables $X_i \in (0,1)$ such that $\sum_{i=1}^{n}X_i = 1$ and let the sequence of $x_i$ be a random realization of $\bar{X}$. The distribution of $\bar{X}...
1
vote
0answers
36 views

Trying to show that this is a $2$-dimensional surface

I have been working on some problems within multivariate analysis, and I am trying to go about proving that given the set $$B = \{(u,v) \in [0,1] \times [0,1] | u < v\}$$ and the set $$Q = \{(1+uv,...
1
vote
1answer
34 views

projection of an ellipsoid on XY plane

The equation of an ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z')....
0
votes
0answers
17 views

How can I see that the image of an open subset of an open set under a function is an open set?

Let A an open set of ℝⁿ and ƒ:A→ℝᵐ of class C'(A). If for some P₀∈A, ƒ'(P₀) is surjective, prove that exists δ>0 such that B$_{δ}$(P₀)⊂A and for all open subset Ω⊂B$_{δ}$(P₀) it holds that ƒ(Ω) is ...