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).
22,154 questions
0
votes
0answers
8 views
Parametric equation for the tangetn curve
Find parametric equations for the tangent line to the curve with the
parametric equations $x=t, y=t^2, z=t^3$ at the point $(1, 1, 1)$.
For the Solution I know the method of solving it. But have a ...
0
votes
0answers
24 views
What formulas of differential geometry am I missing?
I was reading a book on Riemannian analysis and the author assumes some formulas of differential geometry, which may be basic but I have a lack of knowledge on those.
Specifically:
$\int \delta (a(x)...
-1
votes
0answers
19 views
Derivative of the expression [on hold]
How to the equations below and mathematically derive the expression for the rate of change of the dependent variable with respect to its independent variable (s).
$𝑞=20𝑥^{0.6}𝑦^{0.2}𝑧^{0.3}$
0
votes
0answers
27 views
Proof based on orthogonal matrix
I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix.
Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$
How should I approach this proof(...
0
votes
0answers
13 views
Partial derivatives of a composed function $F = g \circ \mathbf{f}$
I am asked to find the formulas for the partial derivatives of a compound function $F := g \circ \mathbf{f}$, with (my answer is in red):
$\mathbf{f}(x,y,z)$ and $g(u,v)$:
$$ \color{red}{\frac{\...
2
votes
0answers
12 views
Using the Fundamental Calculus Theorem for two variables to prove smoothness.
There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters:
$Pr(v)=\int_{G(v)} dP(v)$,
where $G (v)$ is the "goal" region, the probability of getting ...
0
votes
1answer
16 views
Does there exists a linear function $f$ s.t. $\forall$ vectors $a,b,c\in \Bbb R^n, (a-b)\cdot c = a\cdot f(a,b,c)$?
Does any linear function $f$ exist such that $\forall a, b, c \in \mathbb{R}^n: (a-b)\cdot c=a\cdot f(a, b, c)$?
And if so, how to find such function?
"$\cdot$" denotes the scalar product. The ...
-1
votes
0answers
19 views
Triple integral : volume of $4x^2+3y^2=z^2+2$
I need to find the volume of this region $4x^2+3y^2=z^2+2$ for $|z|\le 2$
. It's an elliptic hyperboloid .
$4x^2+3y^2=z^2+2=\sigma$
$\frac{x^2}{\sigma/4}+\frac{y^2}{\sigma/3}=1$
volume = $\...
0
votes
3answers
52 views
How to find the jacobian of the following?
I am stuck with the following problem that says :
If $u_r=\frac{x_r}{\sqrt{1-x_1^2-x_2^2-x_3^2 \cdot \cdot \cdot-x_n^2}}$ where $r=1,2,3,\cdot \cdot \cdot ,n$, then prove that the jacobian of $...
0
votes
0answers
17 views
Taylor's formula and multi-indices
Context: proving Taylor's formula as done in Saint Raymond's Elementary Introduction to the Theory of Pseudodifferential Operators
We want to show that, for a $C^k(\mathbb{R}^n)$ function $u$, for ...
4
votes
5answers
87 views
Evaluating $\lim_{(x,y)\to(0,0)}\frac{x^2+y^2}{\sin^2y+\ln(1+x^2)}$
$$\lim_{(x,y)\to(0,0)}\frac{x^2+y^2}{\sin^2y+\ln(1+x^2)}$$
If I use a specific path I know I can use Cauchy Theorem to get a number, but how do I prove this for all paths? Thank you!
0
votes
0answers
20 views
Multivariable Taylor Expansion of $f(\mathbf{x}+s\mathbf{p})$
I was going through some unconstrained optimization class notes and in order to prove the $1^{st}$ Order Necessary Condition for Optimality:
$\mathbf{x}^{*}$ is local or global optimal $\Rightarrow$ $...
0
votes
0answers
17 views
Multi-dimensional integration by parts
Here's my problem: Given $X=(X_1,X_2)$ is a centered Gaussian random vector, i.e., $X\sim \mathcal{N}(0,C)$ and the density of $X$ is given by
$$
p(x)=p(x_1,x_2)=\frac{1}{(2\pi)^{m/2}|C|^{1/2}}\exp\{...
0
votes
1answer
34 views
How do I find the infimum of a function using Calculus III techniques?
Let $f(x, y, z) = \frac{x + y + z}{2} - \sqrt{xyz}$. Find the infimum of the function where the domain is restricted to the first quadrant.
There are techniques in Calculus III involving the hessian ...
2
votes
2answers
40 views
Computing this limit: $ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$
If $f(x,y) \in \mathbb{R^2}$ and $g(x) \in \mathbb{R}$. Assuming
$\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$
$$$$
Can we do the following:
$$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y}...
0
votes
0answers
11 views
Supermodularity and increasing differences: Am I correct?
For background, I have been given the following definition of increasing differences:
$$
\text{A function }F : X \times T \rightarrow \mathbb{R}\text{ has increasing differences in }(x,t)\text{ if and ...
0
votes
3answers
39 views
Limit of $\frac{\sin(xy^2)}{xy}$
I found this interesting problem on calculating the limit of $\frac{\sin(xy^2)}{xy}$ on the positive coordinate axes $x$ and $y$. That is, compute the limit on the points $(x_0, 0)$ and $(0,y_0)$ when ...
1
vote
1answer
34 views
winding number of paths
Let $c:[0,1]\to\mathbb{R}^2\backslash\{\mathbf{0}\}$ be a closed path with winding number $k$. Let $\tilde{c}=\rho(t)c(t)$, where $\rho:[0,1]\to(0,\infty)$ is function satisfying $\rho(0)=\rho(1)$. ...
3
votes
4answers
200 views
Higher dimensional volume using triple integral
As a normal single variable integral is used to find an area under a certain region below a 2d-curve and double integral are used to find the volume under a 3d-curve, I used to think triple integrals ...
0
votes
1answer
16 views
Why each component of gradient which is slope of the curve in itself while keeping other variables constant gives us slope of curve?
My doubt is suppose we assume a 3D space with 2D surface in it given by some function z = f(x,y). Then each component of the gradient is geometrically the slope of the tangent at f on either x-z or y-...
0
votes
1answer
37 views
Minimum in a non-linear system
I have the linear system:
$$\begin{cases}\dot{x}=y\\
\dot{y}=-ay+x-x^3\end{cases}$$
where $a\geq 0$.
I want to prove that this dynamical system has two minimum.
I found the 3 equilibrium points $(...
4
votes
2answers
36 views
Differentation of vector
in the below equation, $\mathbf w$ is a vector with components $w_0$ and $w_1$.
$x^{(i)}$ and $y^{(i)}$ are constants.
how to differentiate $j(\mathbf w)$ w.r.t. $w_0$ and $w_1$
$j(\mathbf w) = \...
2
votes
1answer
54 views
For what values of $a$ and $b$ is the function $\frac{x^ay^b}{x^2+y^2}$ continuous at $(0,0)$?
I have the function $$f(x,y)=\begin{cases}\dfrac{x^ay^b}{x^2+y^2} &(x,y)\neq(0,0)\\ 0 &(x,y)=(0,0) \end{cases}$$ I am trying to figure out what constants $a$ and $b$ will make the function ...
2
votes
1answer
40 views
Calculating iterated integral using Sagemath
I would like to calculate the integral over the following domain (with order $x,y,z$) using Sagemath
$$0 \le z \le 3, \max\{ 0,\frac{z-1}{2}\} \le y \le 1, \max \{y,z-2y\} \le x \le 1. \tag{1} \label{...
2
votes
1answer
51 views
Why is $\, \int_0^1 \{ \int_0^1 \frac{x-y}{(x+y)^3} \, dy \} \, dx \, \neq \,\int_0^1 \{ \int_0^1 \frac{x-y}{(x+y)^3} \, dx \} \, dy \,$?
We know Double Integrals of a function depend only on (i)Region of Integration and (ii)Function, and not on its order of integration.
In this case :
(i)Region of integration is a square (ABCD with ...
0
votes
1answer
46 views
Re-writing $\int_{0}^{1} \int_{0}^{x}\int_{0}^{x+2y} \mathrm dz\,\mathrm dy\,\mathrm dx$ with the $x$ and $y$ directions first
Can you guys tell me, with reasoning, how to write this triple integral in $x$ and $y$ direction first? I found it very hard to do through graphing.
$$\int_{0}^{1} \int_{0}^{x}\int_{0}^{x+2y} \...
1
vote
1answer
33 views
Derivative of a composite function
There are two functions f(x) and g(x):
f(x) and g(x)
I need to differentiate:
(a) g ∘ f using the chain rule
(b) h, where h = g ∘ f
I found the partial derivatives of f and g with respect to ...
3
votes
1answer
38 views
Vector differential equation
In electromagnetism we often have a perpendicular constant magnetic field causing a charge to move in a circle. My question is, how do we formally solve this differential equation which involves a ...
0
votes
0answers
31 views
Finding a multivariable function using it's antiderivative
If I'm given: $\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$.
Can I do the following steps:
$$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$$
$$ \lim_{y\to0} \frac{\partial{f(x,y)}}{...
1
vote
0answers
29 views
Is there no similar proof to change of variables for double integrals as there is for a single integral?
The single variable change of variable theorem states that, for any function $\varphi$ with integrable derivative, and a continuous function $f$:
$$\int_{\varphi(a)}^{\varphi(b)}f(x)dx=\int_a^bf(\...
-1
votes
0answers
12 views
Help with proving Angular Momentum Balance and Symmetry of stress tensor [on hold]
I can't for the life of me figure this out. I have pored through two textbooks on transport phenomena, and this isn't making any sense. Nothing online seems to have a good suggestion for how to solve ...
0
votes
0answers
17 views
how to calculate $\int_C~ydx + zdy + xdz$, where $C$ is oriented counterclockwise, as viewed from above. ??
Let $C$ be the boundary of the portion of ${z = x
^2 + y
^2}$ below the plane
{$6x + 2y + z = 20$}. Find
$\int_C~ydx + zdy + xdz$, where $C$ is oriented
counterclockwise, as viewed from ...
5
votes
0answers
58 views
Unique solution for a PDE
Consider the equation $$xu_x+yu_y=\frac{1}{\cos(u)}$$
with the initial condition $$u(s^2,\sin(s))=0.$$
If we use the methods of characteristics we can obtain: $$x(s,t)=s^2e^t$$
$$y(s,t)=e^t\ \sin(s)$$...
0
votes
0answers
29 views
Help proving this map is closed
Define the following equivalence relation on $\mathbb{R}^2$: $(x,y)\sim(x',y')$ iff there is $n\in \mathbb{Z}:(x',y')=(x+n,(-1)^ny)$. Let be $E=\mathbb{R}^2/\sim$ the quotient space, and $q:\mathbb{R}^...
1
vote
0answers
27 views
Finding the canonical form of a PDE
Consider the equation
$$u_{xx} + yu_{yy} + \frac {1}{2} u_y = 0$$
If we calculate the discriminant of this equation we find that:
$$\delta (x,y) = -y$$
So for $y<0$ it is a hyperbolic equation ...
0
votes
0answers
7 views
Canonical form of PDE and the idea behind the classification of second-order linear PDEs
I am so sorry for asking such a question but I’m totally new to this subject and I’m confused so I’m trying to use your help and experience.
Consider the equation
$$(1+\ sinx)u_{xx}+ 2\ cos(x)u_{xy} +...
0
votes
2answers
51 views
Find if the function $\frac{(1-2xy)}{(x^2 +y^2)}$ has a max or min value for $(x,y)=/=(0,0)$
Does the function $\frac{1-2xy}{x^2 +y^2}$ have a max or min value for $(x,y)=/=0$?
What I've tried so far is to take the the partial derivatives:
$$\frac{\partial f}{\partial x} = \frac{2(-x+x^2*y ...
0
votes
1answer
47 views
Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the condition $ax^2+by^2+cz^2 =1\;\;$ and $\;\;lz+my+ny=0$
Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the
condition $ax^2+by^2+cz^2 =1\;\;$ and $\;\;lz+my+ny=0$ and interpret
the result geometrically
I started with Lagrange's ...
1
vote
0answers
20 views
Calculating Parametric Path of Particle From Force Field
This question arose when I was doing some work with line integrals over force fields. In these questions, the reader is always given a parametrization of the particle's path (in terms of time) and ...
0
votes
1answer
49 views
Good textbook for questions/answers?
Does anyone know of any good textbooks which contain a lot of exercises and solutions? A lot have exercises, and that's useful but I really would prefer having solutions if I'm going to be doing 100+...
1
vote
2answers
51 views
Find the limit as $(x,y)$ approaches $(0,0)$ of $\frac{\sin(x)(1-\cos(y))}{6xy^{2}}$
I used the path $y=x$ and got that the limit equals $12$. How do I manipulate the original expression to prove that the limit actually equals $12$ for all paths? Thank you!
1
vote
0answers
22 views
Domain for which there exists a unique solution for PDE
I have solved the PDE:
\begin{cases}
xu_x+yu_y=\sec u \\
u(s^2,\sin s)=0
\end{cases}
using the method of characteristics, and if I’m correct we have:
\begin{align}
x(s,t)&=s^2e^t \\
y(s,t)&=e^...
1
vote
2answers
50 views
General solution of a PDE- Lagrange or Characteristics method
I am trying to find the general solution of the PDE:
$$xu_x + (1+y)u_y= x(1+y)+xu$$
If the initial condition is $$u(x,6x-1)=\phi(x)$$
then what is the necessary condition for $\phi$ that guarantees ...
0
votes
1answer
32 views
Prove the discontinuity of $f$ at origin.
Prove the discontinuity of $f $ at origin $ f(x,y)=\begin{cases}\frac{x^2-xy}{x+y},&\text{if $(x,y)\neq(0,0)$} \\ 0,&(x,y) \>\ = (0,0)\end{cases}$ .
Also show that $f $ has partial
...
0
votes
0answers
18 views
How to prove that a parametric curve is the intersection of 2 surfaces
So I've been working on a way to prove rigorously that a parametric curve is the intersection of two surfaces but I'm unsure of how to show it
Question
Show that the curve with parametric equations ...
0
votes
0answers
13 views
Show $\int_{dM} w=\int_M dw$, use $d(w|_M)$ not $w\in \Bbb A(\Bbb R^3)$ , $w|_M \in \Bbb A(M)$.
Let $w \in \Bbb A^1(\Bbb R^3)$, $w=xzdy-yzdx$, $M=\{z=f(x^2+y^2\} over $x^2+y^2 \leq \Bbb R^2$.
Illustrate Stoke's theorem with $(M,w|M)$.
For easier use $f(x)=x$.
Show $\int_{dM} w=\int_M dw$, use $...
0
votes
2answers
42 views
Help with Multivariable Delta-Epsilon Proof for $(x,y)\to (0,\pi/2)$ of $\sin(x+y)=1$ [on hold]
Help with Multivariable Delta-Epsilon Proof for
$$\lim_{(x,y)\to (0,\frac{\pi}{2})}\sin(x+y)=1.$$
0
votes
1answer
22 views
Understanding jacobian matrix
I have a function $f(x,y,z) = x^2+y^2-z^2$, then $df_{(x,y,z)}=(2x,2y,-2z)$. Now the statement is if $a\neq 0$ and $f(x,y,z) = a$ then $df_{(x,y,z)}\neq 0$ and I can't understand why $df_{(x,y,z)}\neq ...
2
votes
0answers
18 views
Easy Integration by Parts in Spherical Coordinates
I am trying to use the integration by parts formula in spherical coordinates
$$\int_\Omega \frac{\partial u}{\partial x_i} v d \Omega = \int_\Gamma u v \nu_i d\Gamma-\int_\Omega u \frac{\partial v}{\...
-1
votes
1answer
26 views
Find the volume integral of the function $f = x^2 + y^2 + z^2$ over:
a) the region given by $0 \leq x \leq 1, 1 \leq y \leq 2, 0 \leq z \leq 3$,
b) the region inside the sphere of radius $R$ centred on the origin
For a) the first thing I've done is convert the ...
