Questions tagged [multivariable-calculus]
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).
22,862 questions
0
votes
0answers
7 views
Weak Solutions Diffusion Equation
Let $\Omega$ be a bounded open set of $\mathbb{R}^n$, with smooth boundary $\Gamma$. Consider an evolution equation in the cylinder $Q=\Omega\times ]0,\,T[\quad T>0, \text{finite}$. Set $\sum=\...
0
votes
0answers
10 views
How to know whether to choose the x-bound or the y-bound for this triple integral
In my textbook for calculus 3, I have been working on example of the triple integral. Though I do know polar, cylindrical, spherical coordinates, this section of the book expects you to work with ...
0
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0answers
16 views
Change of variables in triple integral
Let $D$ be the region in $xyz-$space defined by inequalities $1 \le x \le 2, 0 \le xy \le 2 $ and $0\le z \le 1$. I want to evaluate $\displaystyle \int\int\int_D (x^2y + 3xyz) \text{dxdydz}$ by ...
0
votes
1answer
22 views
Transform coordinate system for the gradient of a function at a specific x, y, z value.
I have the gradient of a function $f(x,y,z)$ at a specific value of $x, y,$ and $z$ in the vector form:
$$
\nabla f(x,y,z)\Bigr|_{\substack{x=x_1\\y=y_1\\z=z_1}} =\begin{pmatrix}
\frac{\partial{f}}{\...
0
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0answers
14 views
A vector calculus formula
Let $A, B$ be vector fields in $\mathbb R^3$. We have
$$
\text{curl}\bigl((A\cdot \nabla)B\bigr)=(A\cdot \nabla)\text{curl}B
-((\text{curl}A)\cdot \nabla)B+R(A,B).
$$
I know that $R(A,A)=0$ and I ...
0
votes
1answer
16 views
One to one relation between functions that coincide in certain values
I was reading a proof of multivariable calculus that suggested the following property might be true:
Given two functions $f(x)$ and $g(x)$ in $\mathbb{R}^n$, if $\forall x_1,x_2$ $f(x_1)=f(x_2)$ $\...
2
votes
2answers
45 views
Finding the global minimum of $\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$ having just the local minimum.
In order to calculate the values of $a$ and $b$ such we get the minimum possible for:
$$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$
I got the help of @TheSimpliFire among others to ...
1
vote
3answers
38 views
Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$
Excercise
Let $X, Y$ be random variables such that their joint density function is defined by:
$f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$.
Find the distribution of Z defined as:
$Z=\frac{X+Y}{2}...
0
votes
1answer
22 views
Is Schwarz's Theorem valid for higher order derivatives?
I know that $\frac {\partial^2 f}{\partial x \partial y} = \frac {\partial^2 f}{\partial y \partial x}$ when $\frac {\partial f}{\partial x}$, $\frac {\partial f}{\partial y}$ exist and are continuous,...
6
votes
2answers
168 views
How to compute a Jacobian using polar coordinates?
Consider the transformation $F$ of $\mathbb R^2\setminus\{(0,0)\}$ onto itself defined as
$$
F(x, y):=\left( \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}\right).$$
Its Jacobian matrix is
$$\tag{1}
\begin{...
0
votes
0answers
14 views
How to convert spherical co-ordinates of a vector field to cartesian co-ordinates :
v$(r,φ,θ) = (r cos2 θ)$r$ − (rcosθsinθ)$θ$ + 3r$φ, where r, θ and φ are the unit spherical vectors.
I was trying to calculate the line integral of the function along the path described in the ...
1
vote
1answer
28 views
Existence of a function with $||grad f||>\epsilon$
I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
0
votes
1answer
32 views
Understanding the domain of the triple integral for $f(x,y,z)=x+2y+z^2$
So, I am having trouble (again) with the domain for a triple integral of a function, bounded by the paraboloid $2y^2=x$ and the $x+2y+z=4$ and $z=0$ planes
I have tried to guess the bounds for x,y ...
0
votes
2answers
19 views
Application of the Divergence Theorem with change of variable
Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
0
votes
0answers
13 views
Partial differential Equation uniqueness
Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following partial differential Equation;
\begin{align}
\nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \...
0
votes
0answers
13 views
Reference for the multivariate Leibniz rule of many factors
I'm looking for a reference (a book/article) with a formula to
$$
\frac{ \partial ^ k }{ \partial x_1^{k_1} ...
\partial x_n^{k_n} } f_1(x) ... f_m(x) ,
$$
where $k=k_1+...+k_n$, $x=(...
2
votes
0answers
30 views
If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$
If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$.
...
1
vote
1answer
57 views
Partial Derivative Disambiguation
There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
3
votes
1answer
25 views
ODE in $\mathbb{R}^n$ defined by the gradient of a function
I'm studying for an exam and I got stuck in this question:
Let $x: I \to \mathbb{R}^n$ be a differentiable parametrized curve (I is an interval) in $\mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$...
1
vote
0answers
25 views
Chain rule for a transformation from $\mathbb{C}$ or $\mathbb{C}^2$ to $\mathbb{R}^2$
Note that, for $z=x+iy\in \mathbb{C},$
$x=\frac{z+ \bar z}{2}\in \mathbb{R}~\hbox{and}~y=\frac{z-\bar z}{2i}=-\frac{i}{2}(z-\bar z)\in \mathbb{R}.$
I was wondering if the following is true:
$$
\frac{\...
0
votes
1answer
43 views
Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.
Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
0
votes
1answer
19 views
Finding the curve of intersection of a cylinder and cone
I have a cone $x^2 + y^2 -z^2 =0$ and a cylinder $ x^2 +y^2 -2ax =0$. Together they look like this.
If one were to project the intersection onto the xy plane, the curve given by this intersection ...
3
votes
2answers
21 views
Showing a mapping is bijective if and only if a matrix is invertible
Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and
$x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping
$\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by
...
0
votes
1answer
24 views
Understanding the domain of the triple integral for $f(x,y,z)=x^2+y^2$
So, I am having trouble with the domain for the triple integral of $f(x,y,z)=x^2+y^2$, bounded by the paraboloid $x^2+y^2=2z$ and the $z=4$ plane
I am currently trying to project it on the XY axis, ...
0
votes
1answer
20 views
Checking saddle point or not - using rules of 'Fundamental Theorem of Calculus'
True or false
For the function, $f(x,y)=\int_{2x}^{-y+2{x^2}}e^{-{t^2}}dt$ $(0,0)$ points is the saddle point.
I can do it the long way by solving the integral first but I believe there is a way to ...
0
votes
0answers
16 views
Proving $\arg$ restricted to an open subset of $S^1$ is smooth
Let $U$ be an open subset of $S^1\subset \mathbb{R}^2$.
Define $\theta:U\to \mathbb{R}$ by $\theta(x,y)=\arg(x+iy)$ where $\arg$ is the principal argument. I want to prove that $\theta$ is smooth (i.e....
3
votes
0answers
10 views
Can we perturb a map to have distinct singular values?
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
-2
votes
4answers
62 views
What's “limit doesn't exist” multiplied by limit that equals zero?
OK, we're having a strong discussion just a day before our Calculus exam. The problem's next: To check if this function is continuous:
$$\frac{y^2\,\sin x}{x^2 + y^2}$$ at (0,0).
We get DNE $\cdot 0$, ...
0
votes
1answer
44 views
Find the area between $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$
So, I have to calculate an integral with a domain limited by two functions: $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$ , where $a>0$
The issue here is that I cannot wrap my head around what the ...
0
votes
1answer
16 views
How does Green's theorem imply the divergence theorem in the plane?
Both Green's theorem and Stokes' theorem involve the integral of a curl and it is easy to see that Green's theorem is a planar version of Stokes' theorem.
However, the divergence theorem involves the ...
0
votes
0answers
22 views
Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.
If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by:
$f(x,y)=
\begin{cases}
(x,y-x^{2}) & if & x^{2} \leq y \\
(x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
1
vote
1answer
31 views
Picture flow of ODE
Consider the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\
\Phi(x,0) = x, \quad x \in \mathbb{R}^2
\end{cases}$$
Suppose that the flow $\Phi$ ...
1
vote
0answers
15 views
Estimating multivariate random walk plus noise model using OLS
I'm currently working on replicating a scientific paper for practice in which they estimate a multivariate random walk plus noise model, apparently using OLS. I have no clue however how they would do ...
0
votes
0answers
7 views
Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$
Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$
So I parametrized $S$ by $\sigma(r,\theta)=(r\cos \theta,r\sin \theta,r^2)$ where $0\leq r\leq \sqrt{5}$ and $0\...
0
votes
0answers
14 views
Will lagrange multiplier method find all stationary points, or just minima and maxima?
Let $f,g : \mathbb{R}^n \to \mathbb{R}$ be smooth functions and $U_c = g^{-1}(c)$ for each $c \in \mathbb{R}$. For each $\lambda \in \mathbb{R}\setminus\{0\}$, consider the equations
$$
\nabla g = 0 \...
2
votes
1answer
19 views
Why do we take magnitude into account when calculating the directional derivative?
Given that the directional derivative is defined formally as:
$$
\nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|}
$$
...
0
votes
0answers
29 views
Triple integral to calculate the volume of pyramid vs it's formula
It is well known the formula to calculate the volume of a pyramid:
$V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the base to the apex.
However I need to ...
0
votes
1answer
28 views
Question about the gradient of a composite function
I am new to calculus and am trying to work out the following question, with no success so far… Any feedback would be great!
Within function $f(x,y)$, variable $y$ is a function of $(x,z)$, in other ...
0
votes
1answer
21 views
Question about gradient in a complex function
I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated!
The function $f(x,y,z)$ is differentiable at all points, and satisfies $f(x,y,2x^2+y^...
0
votes
2answers
23 views
Expand the given function in an appropriate cosine or sine series. (Fourier series)
$$f(x) = \begin{cases} x-1, \quad& -\pi < x <0 \\ x+1, & 0 \leq x \leq \pi \end{cases}$$
This is odd so we expand with an odd series
$$b_n = \frac{2}{\pi} \int_{0}^{\pi} (x+1) \sin(nx) ...
1
vote
1answer
36 views
Fourier series confusion
Find the Fourier series of
Question 1
$$
f(x) =\begin{cases}
0&&\text{for $-1 < x < 0$}\\\\
x&&\text{for $0 \leq x \leq 1$}
\end{cases}
$$
and
Question 2
$$f(x) = x + \pi,\;...
3
votes
2answers
27 views
Prove the vector calculus identity $\frac{1}{2}\mathbf{\nabla(\lVert u \rVert ^2) = (u \cdot \nabla)u+u \times (\nabla \times u )}$
My attempt:
Consider the first component of both sides.
$$LHS=\frac{1}{2}\frac{\partial}{\partial x_1}(u_1^2+u_2^2+u_3^2)=u_1 \frac{\partial u_1}{\partial x_1}+u_2 \frac{\partial u_2}{\partial x_1} + ...
0
votes
1answer
38 views
Selling inventory while maximizing the present value of profit
A company owns an inventory of $100$ units of a good. It must sell the entire inventory over the next three periods. The profit function for sales within any one period is
$$\pi(x_t) = 50x_t − ...
0
votes
1answer
25 views
Showing that $f: D_1(0) \subset \mathbb{R}^2 \to \mathbb{R}, f(x) = \frac{1}{1- \|x\| }$ is continuous
I understand that I need to show that for all $x_0 \in D_1(0)$,
$$\lim_{x\to x_0} \frac{1}{1- \|x\| } = \frac{1}{1- \|x_0\| }$$
But I have trouble bounding the denominator,
$$\bigg| \frac{1}{1- \|x\...
0
votes
1answer
37 views
Integration on Differential Forms.
So I know that divergence and curl of a vector field $F$ can be related to a differential form $\alpha$ by div $ F = \star d \star \alpha$ and curl $ \cdot F = \star d \alpha $ , where $\star$ is the ...
11
votes
2answers
173 views
Find $a$ and $b$ for which $\int_{0}^{1}( ax+b+\frac{1}{1+x^{2}} )^{2}\,dx$ takes its minimum possible value.
Calculate for which values $a$ and $b$ the integral
$$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$
takes its minimum possible value?
For being honest I'm not sure how to ...
0
votes
0answers
25 views
Why can't these two mappings be bijective?
Let $\phi : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuously differentiable function and define the mapping $\mathbf{F} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by
$$\mathbf{F}(x, y) = (\...
1
vote
0answers
42 views
Total differential and overestimation
Consider the function $f(x,y)=10-(x^2+y^2)$ at (1,1) with dx=2 and dy=3
The function value at (1,1) is 8.
Computing the change in the function by taking a first order total differential gives us $-(...
1
vote
1answer
22 views
Double integral to calculate using Polar coordinate system?
I've met a double integral which seems to be calcultated in polar system.
$\iint_D \frac {y} {x^2+y^2} dxdy$ , where $D$ is the region bounded by the following conditions:
$-2y \le x^2+y^2 \le -4y$;
...
0
votes
1answer
24 views
Why is the normal vector different in cartesian coordinates vs. spherical coordinates?
Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere.
Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...
