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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).

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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 ...
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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)...
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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). $𝑞=20𝑥^{0.6}𝑦^{0.2}𝑧^{0.3}$
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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(...
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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{\...
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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 ...
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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 ...
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0answers
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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 = $\...
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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 $...
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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 ...
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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!
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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 $\Rightarrow$ $...
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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\{...
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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 ...
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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}...
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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 ...
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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 ...
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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 $\rho(0)=\rho(1)$. ...
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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 ...
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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-...
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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 $(...
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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) = \...
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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
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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{...
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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 ...
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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} \...
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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 ...
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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 ...
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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)}}{...
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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(\...
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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 ...
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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 ...
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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: $$x(s,t)=s^2e^t$$ $$y(s,t)=e^t\ \sin(s)$$...
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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}^...
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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 ...
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0answers
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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} +...
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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 ...
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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 ...
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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 ...
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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+...
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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!
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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 ...
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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
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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 $...
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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
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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}{\...
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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 ...