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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1
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1answer
11 views

Evaluating a limit by applying differentiability

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is differentiable at the point $(x_0,y_0)$ then $$ \lim_{t\to 0} \frac{f(x_0+tx,y_0+ty)-f(x_0,y_0)}{t}=xf_x(x_0,y_0) + yf_y(x_0,y_0) $$ I know if the ...
1
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1answer
28 views

Find the maximum and minimum of $f(x,y)=xy-y+x-1$ on the set $x^2+y^2=2$

So I started the problem by first finding the critical points using the partial derivatives, which turns out that there is only 1 critical point at $(1,-1)$ where $f(1,-1)=0$ Then I know I must look ...
0
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0answers
8 views

Lagrange Multipliers for Implicit Functions

How can I find the minimum / maximum of a function for one variable defined implicitly (f(x, y, z) = c) with a constraint g(x, y) = c on the domain? For example, say you wanted to minimize for z: ...
1
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0answers
9 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
0
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1answer
15 views

Derivative of f(x, y, z(x, y))

I have computed the derivatives to be: $dw/ds = (-5y+5z)(t) + (-5x-2z)(e^{st}(t))$ $dw/dt = (-5y+5z)(s) + (-5x-2z)(e^{st}(s)) + (-2y + 5x)(2t)$ I calculated the first answer by evaluating dw/ds ...
2
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1answer
301 views

if the curvature is constant and positive, then it is on the circunference

I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
1
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0answers
20 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
0
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0answers
7 views

Multivariable limit involving trig functions

$$ \lim_{\vec{x}\to \vec{0}} \frac{|\cos(x+y)-1|}{\sqrt{x^2+y^2}} $$ I want to evaluate this limit by using the squeeze theorem. I know $$ |\cos(x+y)-1| \leq (x+y)^2 $$ but I can't figure out a way ...
0
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0answers
18 views

Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$

I'm trying to solve this problem: Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$ First, I founded the gradient: $\nabla f(x,y,z)= (2x\cos{z}, ...
7
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1answer
206 views
+100

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
4
votes
1answer
83 views

To find the volume dilation, integrate the determinant of the Jacobian

On the road toward proving the change of variables theorem in several variables, is there a painless way to show that $$\text{Vol}(\phi(U))=\int_{U}|\text{det}(d\phi)|,$$ where $\phi$ is $C^1$, ...
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0answers
14 views

Are the conditions for multivariable integrability the same?

For a single variable function, the function needs to have a finite number of discontinuities and must be bounded over the interval of integration for it to be Riemann integrable over that interval. ...
0
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1answer
17 views

Finding minimum value of a directional derivative(Multivariate Calculus)

Let $f(x, y) = x ^2 e^{−y^2}$ and $v = (1, 1)$. Find all points $(x, y)$ where $|Dvf(x, y)|$ has its minimum value. What i tried. I know that im order to find the mimimum value of a directional ...
0
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1answer
62 views

Book on calculus of several variables.

I'm an undergraduate student in mathematics and want to study Calculus of several variables currently this semester which involves the use of analysis, vector spaces and linear transformations. Can ...
0
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1answer
57 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
1
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1answer
21 views

Differential equation multivariable solution

I don't understand how I would solve the following problem: Where does the $F(t,y) = -5$ come from? I tried solving it normally, do I create a multivariable function that satisfies $F(t,y) = -5$? ...
0
votes
2answers
14 views

Absolute Max/Min of a function of two variables on a set?

How do you find the absolute maximum/minimum values of the function $f(x,y) = x^2 + y^2 - 8y + 16$ on the given set R where $R = {(x,y): x^2 + y^2 ≤ 25}$ I know the absolute maximum is 81 and ...
0
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3answers
43 views

Show that the limit of $\frac{\sin(x) - \sin(y)}{x+y}$ does not exist.

Trying to show that $$\lim_{(x, y) \to (0, 0)}\dfrac{\sin(x) - \sin(y)}{x+y}$$ does not exist, but I'm having a lot of trouble. So far I've tried splitting the expression into two parts, but ...
-1
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2answers
30 views

Limit 2-variable function [on hold]

Prove that $\displaystyle \lim \limits _{(x,y)\to (0,0)} \frac{x^3y}{x^2+y^4}=0$. Please show it both by $\epsilon$-$\delta$ criterion and sequences definitions.
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1answer
29 views

Line Integral with Arclength Parametrization

Suppose we have an arclength parametrization of a curve in the $xy$-plane given by $x(s)$, $y(s)$ where $0 \leq s \leq L$. We want to integrate a scalar function $f(x,y)$ along this line. Since we are ...
2
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1answer
46 views

help with strange Double Integral: $\iint_E {x\sin(y) \over y}\ \rm{dx\ dy}$

i'm having trouble with this double integral: $$ \iint_E {x\sin(y) \over y}\ \rm{dx\ dy},\ \ \ \ E=\Big\{(x,y) \in \mathbb{R^2} \mid 0<y\le x\ \ \ \land\ \ \ x^2+y^2 \le \pi y\Big\} $$ i've ...
1
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2answers
16 views

Are the plane and the line parallel?

Problem In the picture above are given the coordinates of the points $O(0,0,0)$, $A(6,0,0)$, $C(0,12,0)$, $D(0,0,5)$, $K(0,6,5)$, $L(6,12,4)$, $M(6,8,0)$, $N(0,8,0)$. It seems as if line $KL$ is ...
0
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0answers
10 views

Is it possible for the derivative of a multivariate function to be a function of lesser dimension?

Let's say I have some function $f$ such that $f'(a,b,c,d)$ exists for all $a$, $b$, $c$, and $d$, and that $f(a,b,c,d)$ is dependent upon $a$, $b$, $c$, and $d$. (That is, $f(a,b,c,d)$ can't be ...
0
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1answer
21 views

Misconception about chain rule in multiple variables

Let $z=f(x,y)=e^{x}\sin(xy)$, $x=g(s,t)$ and $y=h(s,t)$. If $k(s,t)=f(g(s,t),h(s,t))$, find $\displaystyle\frac{\partial k}{\partial s}$. Until now, I have found that: $\displaystyle\frac{\partial ...
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0answers
9 views

characterisation of continuity of a function in two variables in polarcoordinates

As the the titel of my question already indicates the question I have is about continuity. I know the "$\epsilon-\delta$ Definition" of continuity and think I have understood it. On the internet I ...
1
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1answer
44 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
1
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3answers
22 views

mulivariable calculus-distance and planes

With 4 points A B C D, how do I find the distance from point D to the plane through A, B, C? This is a rather basic calc question I know but I'm not sure where to start. I imagine I'd probably have to ...
0
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0answers
8 views

Coordinate translation through differential functions

I am trying to write some coordinate translation from 3 continuous "tolerance" functions, $t_x(x,y,z)$, $t_y(x,y,z)$ and $t_z(x,y,z)$ in such a way that: $(x,y,z) \to (x',y',z')$ $(x+t_x(x,y,z),y,z) ...
5
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1answer
79 views

Smoothness in $\mathbb{R}^n$

Embarrasingly simple question, but I got the feeling that I cannot see the forrest for the trees right now: If I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and want to show that it is ...
0
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1answer
24 views

Compute differential of Even function

Let $f:\mathbb{R^n}\rightarrow \mathbb{R} $, If $f$ is a differentiable function and $f(-\vec{x})=f(\vec{x}) ,\forall \vec{x} \in \mathbb{R^n}$. Compute the differential of $f$ in $\vec{0}$. I don't ...
0
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1answer
27 views

Calculate Brieskorn Manifold?

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ ...
2
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0answers
26 views

Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
1
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0answers
21 views

Smooth function on a closed set.

Evans book on PDE's defines for a given open subset $U$ of $\mathbb{R}^{n}$, $C^{k}(\overline{U})=\lbrace u:U\rightarrow \mathbb{R}^{n}$, such that $D^{\alpha}u$ exists and is uniformly continuous ...
2
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3answers
50 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
5
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0answers
144 views
+50

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
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0answers
20 views

3D Vectors and Geometry [on hold]

With the points A = (2, 0, 3), B = (1, 1, 1), C = (0, 1, −1) and D = (1, −1, 2)and using five subtractions, two cross products and one dot products find the distance from point D to the plane through ...
0
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1answer
36 views

On the multivariable Taylor expansion

Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ ...
0
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1answer
13 views

Equation of the Tangent Plane to the Surface

Find the equation of the tangent plane to the surface $$ z= \exp\Big(\frac{3x}{17}\Big)\ln(4y) $$ at the point $(4,1,2.808)$. I got $$ 0.4955512784x+0.506408305y+0.319386581, $$ but this is ...
0
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0answers
8 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
0
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0answers
13 views

Show that if f is strongly differentiable at $x_0$ then it satisfies Lipschitz condition in a neighbourhood of $x_0$.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
2
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1answer
397 views

Calculus on Manifolds (Spivak), problem 2-41(a)

Let $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be differentiable. For each $x \in \mathbb{R}$ define $g_x:\mathbb{R}\to\mathbb{R}$ by $g_x(y) = f(x,y)$. Suppose that for each $x$ there is a unique ...
1
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1answer
29 views

Limit of $\frac{x^2-y^2}{x^2+y^2}\sin(x-3y)$ when $(x,y) \to (0,0)$

Show that $$\lim_{(x,y)\to (0,0)}\frac{x^2-y^2}{x^2+y^2}\sin(x-3y)$$ Does not exists I've tried the traditional patches, but I always find zero as answer. Any hint? Thanks in advance!
0
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1answer
14 views

If partial derivatives of order $m$ are continuous then all partial derivatives of order $\leq m$ are also continuous

I'm working on this problem, I've tried to solve it by using the definition for partial derivative but haven't been able to prove it. Appreciate any help: Prove that if all partial derivatives of ...
0
votes
2answers
56 views

Limit of $\frac{xy^3}{1+y^3}$ when $(x,y) \to (0,0)$

Show that $$\lim_{(x,y)\to (0,0)} \frac{xy^3}{1+y^3} = 0$$ The only way I know of doing it is using squeeze theorem, but I couldn't find any function. Any hint? Thanks!
0
votes
1answer
27 views

Is a multivariable function continuous iff it is continuous with respect to each variable?

I am very uncertain when it comes to understanding the continuity of multivariable functions. If we have, for example, a function $f: \mathbb{R}^{4} \to \mathbb{R}$, and we denote the four variables ...
1
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0answers
9 views

Limit in 2 variables for $r=||(x,y)||$

Show that the function $$f(x,y)=\exp\left(\frac{1}{r^2-1}\right), if \;r<1$$ $$f(x,y)=0, if \; r \geq 1$$ Where $r =||(x,y)||$ is continuous in $R^2$ If $r>1$ or $r<1$ it is clearly ...
-1
votes
0answers
17 views

3D Planes and vectors question [on hold]

Points A = (2, 0, 3), B = (1, 1, 1), C = (0, 1, −1) and D = (1, −1, 2). Using five subtractions, two cross products and one dot products find the distance from point D to the plane through A, B and C. ...
0
votes
1answer
17 views

Surface area of a sphere by cylindrical coordinates

I was resolving a problem of electromagnetism that I needed relating the 1/4 surface area of sphere with electrical field. Well, using spherical coordinates is very easy to do that. Take a look: ...
0
votes
1answer
59 views

Show f is continuous

$$ f(x,y)= \begin{cases} \frac{1}{x^2+y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y)=(0,0) \end{cases} $$ Show that f is a continuous function
1
vote
1answer
24 views

How to set up an integral by these conditions?

I've got these surfaces: $$ z = 0\\ z = 4 - y^2 $$ And a cylinder: $$ x^2+y^2=4 $$ I need to find the volume enclosed by these figures. As far as I understand the limits of integration for $z$ are ...