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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17 views

Derive chain rule for complex functions, from the chain rule for real functions

I'm trying to obtain the chain rule for complex (not necessarily holomorphic) functions $\mathbb{C} \to \mathbb{C}$, using the known chain rule for functions $\mathbb{R}^2 \to \mathbb{R}^2$. The ...
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1answer
13 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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2answers
67 views

Graph connected does not imply $f$ is continuous [on hold]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
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2answers
45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
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0answers
15 views

Problem with center of mass in polar coordinates

When we calculate center of mass using rectangular coordinates, we find the average values in each coordinate. Obviously we can't do this very same thing in polar coordinates: if we integrated a ...
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2answers
22 views

Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
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0answers
14 views

Correctly setting up flux integrals

My question has to do with picking the correct limits for integration. I thought I had it figured out well, but I had an interesting issue with a homework problem. The problems were about Green's ...
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0answers
17 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
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1answer
25 views

Integrability of characteristic function

I have a following questions that I am having trouble. Let $E = \{(\frac{a}{b}, \frac{c}{b}) : a,b,c \in \mathbb{Z}, a \text{ and } b \text{ are relatively prime}\}.$ For what $a \in [1,2]$ is the ...
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1answer
28 views

Which subset of $\mathbb{R}^n$ has zero content

I have a question related to zero content. The questions are For what $n$ there an open subset of $\mathbb{R}^n$ that has zero content? For what $n$ there an unbounded subset of $\mathbb{R}^n$ that ...
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1answer
46 views

Gradient of a vector function

I have a vectorial function $f$, defined on the set of all $n$-dimensional vectors. $f(x) = \log(x^TAx)$, where $\log$ is the natural logarithm, $x^T$ is $x$ transpose and $A$ is a symmetric $n \times ...
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0answers
17 views

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 ...
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0answers
19 views

Change of variables in double integration

I was trying to solve this double integral $\int_{0}^{1}\int_{0}^{y}(1-x)^{59}(y-x)^{27}dxdy$, I could do this by taking binomial expansion but that would be very painful. So a sure thing here is a ...
2
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1answer
14 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
63 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
36 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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0answers
12 views

Rotation Matrix in domain and co-domain basis

I was asked t o derive the rotation matrix counterclockwise with given angle in different domain and co-domain basis. Using what we know from trigonometry I derived the Rotation matrix as: R(Q) = ...
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1answer
9 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 ...
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2answers
44 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
44 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 ...
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61 views
+50

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
59 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
36 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
31 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
33 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 ...
1
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1answer
24 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
34 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
26 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
63 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
19 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
41 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 ...
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0answers
11 views

Taylor's expansion of vector valued functions

Apostol, Calculus, Vol 1 : Let $b$ be a given point in $\mathbb R^n$. Then, if $v$ be any given vector,my textbook defines the taylors expansion of $f$ as : $f(b+v) = f(b) + f~'(b)(v)+ ||~v~|| ...
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1answer
38 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
27 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
20 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
44 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
26 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
31 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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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) ...
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1answer
35 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
42 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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0answers
10 views

Functions x Level Curves Domains

A problem asked me to find the directional vector u which results in a directional derivative dw/ds|u = 0 for the function w = xy+yz+xz at the point p = (1,-1,2). My initial approach to the solution ...
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1answer
24 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} ...
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0answers
27 views

Double Integral with integrand similar to bivariate normal density

I got a double integral like the following, $$\int_{0}^{\infty} \int_{y}^{\infty} xe^{-\frac{(x-by-c)^2}{2a}}ye^{-\frac{(y-e)^2}{2d}}dxdy,$$ where $a$,$b$,$c$,$d$,$e$ are viewed as some other ...
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0answers
8 views

What is the local form of the function at the point of self-intersection with a contour?

I am trying to solve this question : One of the contours (i.e. loci of locations with the same value) of a generic smooth scalar function of the two-dimensional plane is roughly ...
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0answers
6 views

Show the exceptation of a normalized vector [on hold]

Given $n$ $N \times 1$ vectors, $x_1$, $x_2$, ..., $x_n$, which are i.i.d complex Gaussian distributed with zero mean and variance one. Let $z=\sum_{i=1}^n x_i$. Please show that the exceptation of ...
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0answers
37 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
24 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
52 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 ...