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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Multivariable Limits- Non Origin Ones?

I need your help in the following limits: 1) How can I prove the following limit is zero? $$ \lim_{(x,y)\to(1,1)} \frac{\tan(y-1)\sin^2(x-y)}{(x-1)^2+(y-1)^2} \ ? $$ 2) It seems like the following ...
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2answers
58 views

Multivariable Limit Question-Arctan and ln

Can someone help me calculate : $$\lim _{(x,y)\to (1,2)} \frac {\arctan(x+y-3)}{\ln(x+y-2)}?$$ I think substituting $x+y = t $ might help, but I am not sure that doing such a substitution in a ...
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2answers
68 views

Existence of limit in $\mathbb R^2$

I want to prove for a function from $\mathbb{R}^2$ to $\mathbb{R}^2$, its limit at 0 exists. Is it enough to prove that the limit exists and same if we approach $0$ through the all the lines starting ...
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1answer
44 views

understanding simple multivariable integrals in terms of differential forms

I am learning a bit about differential forms: defining differential forms in terms of elementary forms, integrating forms over parametrized domains, etc. I would like to relate this to my previous ...
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1answer
40 views

Am I doing this partial derivative correctly so far?

If $u=\arctan(xy+z)$, where $$x=s^2+t^2,\;y=9re^{st},\;z=r^2st,$$ find the value of $\frac{\partial u}{\partial s}$ when $r=2,s=1,t=0$. Is my attempt so far correct? $$\frac { ∂u }{ ∂s } =\frac ...
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3answers
260 views

Surface area of a plane inside a cone

Determine the surface area of the part of the plane $z=1+x+2y$ which is inside the cone surface $z=\sqrt{2x^{2}+10y^{2}}$.
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1answer
47 views

What are the critical points?

I've found the critical points: $$f(x,y)={ x }^{ 3 }-3x-{ y }^{ 3 }+12y+2\\$$ $$\\ { f }_{ x }=0\\ { x }^{ 2 }-1=0\\ x=\pm 1\\ $$ $$\\ { f }_{ y }=0\\ { y }^{ 2 }-4=0\\ y=\pm 2$$ But I don't know ...
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1answer
80 views

Change of Limits When Changing Order of Integration

When you change your order of integration, you have to change the limits. The way I've been taught is to draw a graph in order to find the new limits. Is there a formula to find the new limits of ...
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1answer
46 views

Show that anecessary and sufficient condition for $x_{p}$ to be tangent to $S^{n}$ at $p$

Please help me! How do I solve this problem? I didnt produce any idea because I didnt understand this topic properly. Thus, please can you explain the solution explicitly? Thank you for help:)
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1answer
78 views

Directional derivative Multivariate

Find the gradient, Hessian and the directional derivative of the following function at [1,1] $$f(x)=x^T \left| \begin{array}{cc} 1 & 2 \\ 4 & 8 \end{array} \right|x + x^T \left| ...
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0answers
99 views

Useful approximation of the pdf

Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
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0answers
59 views

Volume under intersection of 2D Gaussians

Given 2 2D Gaussian functions (probability cdfs, if that matters), is there a direct method to calculate the volume under their intersection? If not, are there any quick ways to approximate it? ...
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1answer
33 views

Identity for $\nabla w .\nabla w$

How can I expand/find an identity for $\nabla w .\nabla w$ so that I get div ($\nabla.$) in the expansion? Thanks!
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0answers
72 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
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1answer
126 views

Stokes theorem problem to find alpha and beta so that I is independent of the choice of S

I have a question that I got half through but can't finish it. If anyone could help I would appreciate it. Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle ...
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2answers
145 views

What does elementary function mean? [duplicate]

I am looking at my double integration example problems and I see a note that says integral of $e^{-x^2}$ is not an elementary function. What does that mean? And why isn't that an elementary function? ...
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3answers
278 views

correcting a mistake in Spivak

Spivak's Calculus on Manifolds asks the reader to prove this (problem 1-8, pp.4-5): If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ ...
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2answers
84 views

Determining the Moment of inertia

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
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1answer
26 views

Maximal region in the cylindrical space

I would like to determine a maximal region in $(r, \theta, z)$- space which maps injectively into $(x,y,z)$-space Thank you
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1answer
217 views

Find a formula for a vector field with given properties

This is the exercise: Give a formula $$F = M(x, y) i+N(x, y)j$$ for the vector fIeld in the plane that has the properties that $$F = 0$$ at $(0,0)$ and that at any other point $(a,b)$, $F$ is tangent ...
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1answer
284 views

Mass of a rectangle

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
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1answer
53 views

Determining Volume by rotation of a plane

a) What is the area of the region R between the graphs of $y= \sin x$ and $y=\sin ^2 x$ for $x \in [0; \frac{\pi}{2}]$ I found $a(R)= 1-\frac{\pi}{4}$ b) Let R be the region from question a). ...
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1answer
141 views

Directional derivative of a scalar field in the direction of fastest increase of another such field

Suppose $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are scalar fields. What expression represents the directional derivative of $f$ in the direction in which $g$ is increasing the fastest?
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1answer
54 views

Solving differential equation for x

I have a field $\phi(x,t)=\sin(t+|x|)(\frac{x}{|x|})$ where x is a point vector and t is the current time. If this field describes the acceleration of a particle at a point in space and time: ...
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1answer
43 views

Two questions about iterated integrals

The two questions are not related. 1) Suppose that $f_k$ is integrable on $[a_k,\;b_k]$ for $k=1,...,n$ and set $R=[a_1,\;b_1]\times...\times[a_n,\;b_n]$. Prove that ...
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2answers
324 views

What does it mean for partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$?

I am reading my text book and I come across a theorem that says: If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$. ...
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0answers
90 views

Directional derivative, tangent vector, double integrals

Let f be differentiable on the domain $D$ and let $\nabla f(x_0, y_0) = (0, 0)$. Then every directional derivative of $f$ at $(x_0, y_0)$ is $0$. Let $C$ be a parametrized curve with ...
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3answers
174 views

Proving that a critical point is a minimum

I am solving a problem, which is "Find the point of the paraboloid $P:=\{(x,y,z)\in\mathbb{R}^3 | x^2+y^2=z\}$ which is the nearest to the point $(1,1,\frac12)$." I have already determined (using ...
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1answer
88 views

$ \int_{C}^{} \frac{ydx-(x+1)dy}{x^2+y^2+2x+1} $ where $C$ is $ |x|+|y|=4 $

Calculate $$ \int_{C}^{} \frac{ydx-(x+1)dy}{x^2+y^2+2x+1} $$ where C is the curve $$ |x|+|y|=4 $$ counterclock-wise, a full revolution. Answer: $$-2\pi$$ So, I've tried to figure this out ...
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6answers
406 views

How do you find the limit of $\frac{4x^4 + 5y^4}{x^2 + y^2}$?

Find the limit of $\frac{4x^4 + 5y^4}{x^2 + y^2}$ as $(x,y)\to (0,0)$. Which method do I use to find the limit of that? I tried paths but the limits all came out to be $0$... (as a side question, ...
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1answer
36 views

parametric integration

Is there a mistake in the bottom of page 5 of this document? INTEGRATION: THE FEYNMAN WAY $$\frac{\partial}{\partial b} e^{be^{ix}}= e^{ix}e^{be^{ix}}$$ instead of $ib e^{be^{ix}}e^{ix}$? thank you ...
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1answer
184 views

Finding image of rectangle under non-linear transformation

For the function $f(x,y)=(xy+x+y,y-x)$ on $\mathbb R^2$, how would I go about finding the image of the region $\{(x,y):1<x,y<2\}$? I am not looking for a solution, just a hint. EDIT: If I ...
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1answer
143 views

How to find a partial derivative of an implicitly defined function at a point

Suppose that the relation $\frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + xz =\frac{7}{2}$ defines $z$ as a function of $x, y$ around the point $(1, 1, 1)$. Find $\frac{dz}{dy}$ at $(1, 1, ...
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243 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
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0answers
74 views

Gradient contradicting dimensions. Find the mistake!

$\nabla diag(X^TX)= diag(\nabla(X^TX))=2diag(X)?$ ?- when $X$ is non symmetric rectangular matrix with real entries. $diag(.)$ denotes a diagonal matrix formed with the diagonal elements being the ...
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2answers
487 views

Finding an explicit inverse on a neighborhood for a function $\mathbb R^2\rightarrow\mathbb R^2$

I have the continuously differentiable function $f(x,y)=(x^3y,xy)$. Its Jacobian is $$\begin{pmatrix}3x^2y&x^3\\y&x\end{pmatrix}.$$ At $(1,1)$, the Jacobian is non-zero, so by the inverse ...
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0answers
54 views

investigating the negative definiteness of a continuous but not differentiable function

I'm stuck at showing whether the following function is negative definite. Consider the following function $V(x_{1},x_{2}):\mathbb{R}^{2}\rightarrow\mathbb{R}$ $$ ...
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1answer
115 views

Cross product with orthonormal basis

Let ${\{u_1,u_2,\ldots,u_n}\}$ be an ortohonormal basis of $\mathbb{R^n}$ and $f,g:\mathbb{R}^n\to\mathbb{R}$ differtinable functions at $p\in\mathbb{R^n}$. If $n=3$ does ...
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1answer
215 views

show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.

If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$ This is a proposition in a book. But I cannot prove this:( ...
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1answer
332 views

Prove Leibniz' rule using three properties

I get stuck after using the first parameter, on how to expand it using the fundemental theorm of calculus and the chain rule. Here is the question: If $\varphi(t) = \int_{g(t)}^{h(t)}f(x,t)dx$, ...
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1answer
181 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
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0answers
458 views

How do you sketch $f(x,y) = 9 - x^2 - 9y^2$ in the $xyz$ plane?

Sketch the graph of $f(x,y) = 9 - x^2 - 9y^2$? How can you do this without using graphing devices? Where do I start? I am really have a hard time sketching these 2 variable functions... Is there a ...
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2answers
66 views

Help with function limit

I need to say whether the limit $$\lim_{(x,y)\to(0,0)}f(x,y),$$ exists, where $$f(x,y)=\frac{x^{2}y^{2}}{|x^{3}|+|y^{3}|}.$$ Can anyone give a hand?
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1answer
61 views

Show that $g$ is continuous using the fact that $f$ is unformily continuous

Working through Advanced Calculus of Several Variables by Edwards. The section is entitled Step Functions and Riemann Sums. Here is the question: Let $A$ and $B$ be contented sets, and $f: A \times ...
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98 views

Chain rule for function of two derivatives

Let $u = f(x, y)$ be a $C^2$-function. Let $x = rcos\theta$ and $y = rsin\theta$. Compute $\frac {d^2u}{dr^2}$ in terms of partial derivatives. Answer: $\frac {d^2u}{dr^2}$ = $\frac ...
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1answer
105 views

Does the total derivative at $(0,0)$ exist: $f(x,y)=\frac{\sin(x^2y^3)}{x^2+(x^2+y^2)^2}$

Does the total derivative of the following function exist at $(0,0)$? $$f(x,y)=\frac{\sin(x^2y^3)}{x^2+(x^2+y^2)^2}$$ Edit: Sorry for the confusion! $f(0,0)=0$, and I proved that $f$ is continuous ...
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2answers
140 views

Multiplying Out Inner Products

If I have a product of the form $(x-s)^tA(x-s)$ where $x$ and $s$ are vectors and $A$ is a matrix, how would I go about multiplying this out? Further, how would I go about taking its derivative?
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1answer
41 views

I have done the second direction of the proof. Hopefully, it is true. Please show my mistakes?

Show that two $C^{∞}$ vector fields $X$ and $Y$ on a manifold $M$ are equal if and only if for every $C^{∞}$ function $f$ on $M$,we have $Xf =Yf$. I have sone one direction of the proof. let $p ∈ ...
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2answers
97 views

Double integral of polar coordinates?

Compute $\int_C (8-\sqrt{x^2 +y^2}) ~ds$ where $C$ is the circle $x^2 + y^2 =4$. Answer: $24\pi$ How is the answer $24\pi$? I converted the integral into a double integral of polar coordinates ...
2
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2answers
443 views

Integral from $0$ to Infinity of $e^{-3x^2}$? [duplicate]

How do you calculate the integral from $0$ to Infinity of $e^{-3x^2}$? I am supposed to use a double integral. Can someone please explain? Thanks in advance.