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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2
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3answers
57 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 ...
0
votes
1answer
10 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 ...
0
votes
0answers
8 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) = ...
1
vote
1answer
8 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 ...
1
vote
0answers
35 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. There is a map $\Delta : X \to L(X,Y)$ continuous at $a$, s.t. ...
1
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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
votes
2answers
36 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 ...
1
vote
4answers
39 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 ...
1
vote
0answers
20 views

How to set up a surface integral on a cylinder cut by planes?

Find the surface area of the piece of the cylinder $x^2 + y^2 = 4$ cut off by the planes $z = 0$ and $y = z$ with $y \ge 0$ using surface integrals. Can someone help me set up this surface integral ...
3
votes
0answers
104 views

Is there a generalization of integration by parts?

In the original integration by part formula there are two functions $u(x)$ and $v(x)$. What if the integral involves another function $w(x)$ as well? Second of all, I know that there is a several ...
0
votes
2answers
29 views

Line integral of vector field [on hold]

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 $b>0$. ...
1
vote
1answer
28 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 ...
1
vote
1answer
23 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 ...
1
vote
1answer
32 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
vote
1answer
24 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 ...
1
vote
1answer
88 views
+50

The path of the shock

Here I am using the shock speed to work out the path the shock takes. I don't understand why we cannot take the value of $u_{-}$ at $t=1/u_0$ i.e $u_{-}=u_0$. and calculate the speed of the ...
0
votes
3answers
32 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?
2
votes
0answers
33 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 ...
2
votes
1answer
25 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
votes
1answer
281 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
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?
0
votes
1answer
39 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
votes
1answer
24 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 ...
0
votes
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~|| ...
2
votes
1answer
37 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
votes
1answer
20 views

parallelepiped change of variables

I can't figure out how to start this problem. Use a triple Integral to find the mass of a parallelepiped generated by the vectors $$<6,1,2>,\ <3,3,9>,\ {\rm and}\ <2,7,3>.$$ We are ...
0
votes
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 $\partial \vec r *A$ ...
1
vote
3answers
42 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
votes
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) = ...
0
votes
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?
0
votes
0answers
12 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
votes
1answer
34 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 ...
0
votes
1answer
40 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 ...
1
vote
2answers
59 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
0
votes
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 ...
1
vote
1answer
29 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 ...
1
vote
1answer
23 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} ...
0
votes
0answers
25 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 ...
0
votes
0answers
6 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 ...
1
vote
5answers
121 views

Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ [closed]

It is given that $4 x^4 + 9 y^4 = 64$. Then what will be the maximum value of $x^2 + y^2$? I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .
3
votes
3answers
49 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
17 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
0
votes
1answer
27 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
votes
2answers
24 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 ...
0
votes
1answer
38 views

Equation that defines multi-dimensional polynomial

In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ . However, now I am dealing with the following two-dimensional ...
1
vote
0answers
19 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 ...
0
votes
0answers
20 views

Find the point Q

Give Q, the point at a distance of 0.1 from P=(4,5) in the direction of v=-i+3j. I used the distance formula, taking Q as x*v but there was no solution for the quadratic equation that came after.
-2
votes
0answers
14 views

Hessian of function of two norm [on hold]

I need to calculate the: ${\nabla ^2}f\left( x \right) = ? $ where $ f(x) = \gamma \left( {a,||x||_2^2} \right)$ and $ \gamma \left( {a,b} \right)$ is the upper bound incomplit gamma function. ...