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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Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
2
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1answer
88 views

Finding surface area S using area of projection of S??

I was going over my calculus textbook and came across a question about surface area. and question is as follows. Let S be a parallelogram not parallel to any of the coordinate planes. Let ...
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0answers
155 views

Calculating electric flux through a sphere (calculus)

This problem comes with two parts, and the reason I am posting here is that they are both supposed to result in the same answer but I am getting two different values. A spherical shell of radius $R$ ...
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2answers
80 views

Volume of solid by Spherical

Trouble setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. Use integration with Spherical coordinates. (Hint: Use two ...
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1answer
21 views

How to find vector $\vec{A}+\vec{B}$ with position vector and displacement vector using different methods

When position vector $\vec{A}$ is $\langle 4, -2, 3\rangle$ and displacement vector $\vec{B}$ is from point $Q(0,4,1)$ to point $R(2,3,-2)$ How am I supposed to find vector $\vec{A}+\vec{B}$ using ...
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0answers
8 views

Does a function need to have a parameter underlying all inputs to be totally differentiable?

I have the following function $$e(p_x,p_y,\bar{U}) = p_x a(p_x,p_y,\bar{U}) + p_y b(p_x,p_y,\bar{U})$$ Can I take the total derivative of this? I am confused about the criteria for something to be ...
2
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1answer
60 views

Having trouble calculating $f_{xx}$ of a “variable-heavy” quotient.

Let $$ f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $$ Compute $f_x (0,0)$, $f_y (0,0)$, $f_{xx} (0,0)$, $f_{xy} (0,0)$, and ...
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3answers
49 views

How do I prove that the limit of $\frac{x^2 y }{x^2 + y^2} = 0$?

How do I prove that $\lim_{(x,y)\to (0,0)} \frac{x^2 y }{x^2 + y^2} = 0$? I can prove this by notifying $x=rcos\theta$ and $y=rsin\theta$, but I remember that it could also be proven by squeeze ...
6
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1answer
255 views

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. Let $f:X\to Y$ and $a\in X$ There is a map $\Delta : X \to L(X,Y)$ continuous at ...
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2answers
112 views

Line integral of conservative vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
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1answer
78 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
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1answer
37 views

Double integral variable change help

I'm having a tough go with this problem. $\iint \frac{x^2}{y^3} dA$ , Integrate using a change of variables over the region defined inside the curves $y=2x,\; y=x,\; y=x^2,\; y=2x^2$ . I graphed it ...
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1answer
49 views

Double Integral Change of variable help

I am having some trouble getting this problem set up, and would appreciate any help. Problem: $\iint \frac{1}{(x+y)^2} dA$. Integrate using change of variables over the region inside the lines ...
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1answer
53 views

How do I simplify a Multivariable expression involving derivatives of logarithms?

I have this expression I got after a lot of calculation: $$\sigma =\frac{d\log\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)}{d\log\left(\frac{ 2 ...
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3answers
45 views

Why would the Jacobian not be zero in this case?

Find the jacobian of the transformation x = u, y = 3uv in the uv plane. Why would $U_y$ not be zero in this case, if the equation U = x contains no mentions of y?
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1answer
48 views

Green's Theorem and limits on y for flux

I'm working through understanding the example provided in the book for the divergence integral. The theorem (Green's): $$ \oint_C = \mathbf{F}\cdot \mathbf{T}ds = ...
2
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3answers
70 views

Question about maximizers and trig

Hi there I have a quick question about the following Consider the simple maximization problem of $$f(x,y)= \frac{x}{1+x^2+y^2}$$ It can be easily seen from analysis of critical points obtained from ...
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1answer
53 views

Two variables Taylor's expansion

I guess that Taylor's expansion about $(0,0)$ is useful for finding value of $\dfrac{\partial^{4n}}{\partial x^{2n}\partial y^{2n}} \left (\dfrac{1}{1+x^2+y^2}\right)(0,0) $. How can it do?
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1answer
99 views

What does it mean for partial derivative to be continuous and how does that imply differentiability?

In order for function to be differentiable at some point, it should be well approximated at that point. I understand that partial derivatives must exist, and that function needs to be continuous, but ...
2
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1answer
49 views

Riemann Integral on $\mathbb{R}^2$

I have the following question. Find a function $f(x,y)$ that is integrable on rectangle $[0,1] \times [0,1]$, such that $g(y) = f(\frac{1}{2}, y)$ is not integrable for $y \in [0,1]$, or prove that ...
2
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1answer
37 views

Level surface undefined

Can a level surface be undefined at some point, even if the original fuction is defined at the same point? example: $w(x,y,z) = xy+yz+xz$ is defined at $p=(1,-1,2).$ Its level surface at $p$ is ...
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1answer
73 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
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3answers
61 views

I need hints to proof $\lim\limits_{(x,y)\to(x_0,y_0)} \frac{\sin f(x,y)}{f(x,y)} = 1 $

Let $f: U\subseteq\mathbb{R^2}\to\mathbb{R}$, $(x_0,y_0)\in U$ and suppose that $\lim\limits_{(x,y)\to(x_0,y_0)} f(x,y) = 0$. Prove that $$\lim\limits_{(x,y)\to(x_0,y_0)} \frac{\sin f(x,y)}{f(x,y)} = ...
3
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2answers
60 views

parametric equation of level curve in three dimensional plane

What is the parametric equation for the tangent plane to the level curve of the function $$w(x,y,z) = xy+yz+xz$$ at the point $(1,-1,2)$? My answer was: $$(x,y,z) = ...
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2answers
49 views

Different results for the same equation

Why does the chart of $xy+yz+xz=-1$, a one sheeted hyperbolid, is different from the chart of $z = -\frac{1}{x+y} - \frac{xy}{x+y}$? Aren't they both the same equation?
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0answers
31 views

How to obtain the line element in cylindrical coordinates, using definition of differential forms

In general, a volume element is a k-form on an K-dimensional manifold. a k-form w on $\mathbb{R}^{n}$ is defined as $w(x) = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}}(x) ...
0
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1answer
38 views

Where is $f_x$ continuous for $f(x,y) = (x^2-y^2)/\sqrt[3]{x^2+y^2}$?

Question Where is $f_x$ continuous for $f(x,y) = (x^2-y^2)/\sqrt[3]{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $f(x,y) = 0$ for $(x,y)=(0,0)$? Issues My attempt to calculate the derivative gave me ...
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1answer
64 views

$f:U \rightarrow \mathbb{R}$, $U$ is an open conected subset of $\mathbb{R}^n$ and $f \in C^1$ need to show that $f$ is $M$ Lipschitz on any compact

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
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1answer
37 views

Integrability of dirichlet function in $\mathbb{R}^3$

Let $d: [0,1] \rightarrow \mathbb{R}$ be the Dirichlet function as follows: $$d(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{R} \backslash \mathbb{Q} ...
2
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0answers
42 views

The Adrian Transformation of a function in $\mathbb{R}^{2}$

Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this: Rotate the area formed by $\int_{-1}^12dx$ around the curve ...
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1answer
44 views

Domain and Range, Vector Calculus

Find the domain and range: $f(x,y) = \sqrt(y^2-x)$ Solution: I found the domain to be $D = \{(x,y)|y^2 \ge x\}$ and range to be $R = \{z| [0, \infty)\}$ Question: I am having difficulty figuring ...
3
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3answers
118 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
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2answers
43 views

Change $\int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx$ to polar coordinates

This is a homework problem, so please do not give more than hints. I must convert \begin{align} \int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx\tag{1} \end{align} to polar ...
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1answer
50 views

What are the images under $f$ of lines parallel to the coordinate axes?

Let $f=(f_1,f_2)$ be the mapping of $\mathbb{R^2}$ into $\mathbb{R^2}$ given by $$ f_1(x,y)=e^x \cos y,\quad f_2(x,y)=e^x \sin y.$$ What are the images under $f$ of lines parallel to the coordinate ...
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0answers
32 views

Bound of integration over the surface area?

Compute the surface area of that portion of the sphere $x^2+y^2+z^2=a^2$ lying within the cylinder $x^2+y^2=ay$ where $a>0$ I first parametize the sphere using spherical coordinate. I think ...
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1answer
34 views

Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor

I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y). Eventually I managed to pull the (presumably) correct ...
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1answer
52 views

Find the directional derivative using $f(x,y,z)=xy+z^2$. [closed]

Find the directional derivative using $f(x,y,z)=xy+z^2$, at the point $(2,3,4)$ in the direction of a vector making an angle of $\frac{3\pi}{4}$ with grad $f(2,3,4)$. PS - I am having trouble ...
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0answers
45 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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0answers
25 views

How do I find $C^1$ mapping with given range

I have a following question. Find a one-to-one $C^1$ mapping $f$ from the first quadrant of the $xy$-plane to the first quadrant of the $uv$-plane such that the region where $x^2 \leq y \leq 2x^2$ ...
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1answer
23 views

Is $Cr^a\hat r$ always a conservative vector field?

Is the vector field $\vec r(r, \theta, \phi) = Cr^a\hat r$, where $C, a\in \Bbb R$ are constants and $r \in \Bbb R^+ \cup \{0\}$ is the radial component, always a conservative vector field? I really ...
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0answers
25 views

Computing the spherical coordinates in n-dimensions [duplicate]

This time I want to compute the Jacobian of the spherical coordinates in n dimensions, so it needs to give me the following result: $$\displaystyle ...
0
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1answer
67 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
6
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1answer
82 views

Find multivariable limit $\frac{x^2y}{x^2+y^3}$

Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a ...
3
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1answer
37 views

Lipschitz condition not satisfied

To show there is no contradiction to existence and uniqueness $\displaystyle\frac{|f(x,u)-f(x,v)|}{|u-v|}= \displaystyle\frac{|x||u^{1/2}-v^{1/2}|}{|u-v|}=\frac{|x|}{u^{1/2}+v^{1/2}}$ I understand ...
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1answer
39 views

maxima minima optimization problem

My problem is Find the greatest and least distance of the surface $6x^2+4xy+3y^2+14z^2=14$ from the origin. I know that mathematical model of problem is $f(x,y,z)=x^2+y^2+z^2$ subject to ...
0
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1answer
100 views

Calculate the integral of $f(x,y,z)=e^z$ over the tetrahedron with vertices $(4,0,0), (0,4,0), (0,0,6)$

How does one setup this integral? Do I simply need an equation for the plane? I want to say this is the answer $\int_{0}^{6}\int_{0}^{4-z}\int_{0}^{4-y-z}(e^z)\space{dx}\space{dy}\space{dz}$ but ...
1
vote
1answer
52 views

Let S be a parallelogram not parallel to any of the coordinate planes. Area of S?

Let S be a parallelogram not parallel to any of the coordinate planes. Let $S_1,S_2,S_3$ denote the areas of the projections of S on the three coordinates planes. Find the area of S in terms of ...
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1answer
35 views

Prove $f$ differentiable: $\|f(v)\| \leq M \cdot \|v\|^{N+1}$

Let $f:\Bbb R^n\to \Bbb R^m$. Assume that there exists $M\in\Bbb R$ positive such that for all $v \in \mathbb R^n$: $$ \|f(v)\|\leq M\cdot \|v\|^{N+1} $$ for some natural number $N\geq 1$. Prove $f$ ...
-1
votes
1answer
41 views

Vector operator

With the help of Vector Operator $\nabla$ and the rules of differentiation and multiplication of vectors, prove the following identities: $$grad (\varphi \psi) = \varphi grad(\psi) + \psi ...
1
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1answer
48 views

Gradient of Vector

What would be the gradient of following vectors in Cartesian Co-ordinate? $$grad (\vec{c} \cdot \vec{r})$$ where $\vec{c}$ is the constant vector and $\vec{r}$ is the radius vector. Thanks.