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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Solution of a Partial Differential Equation

Problem statement Solve $\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=y$ using the change of variables $\left\{\begin{matrix} u=ax^2+y \\ v=x \end{matrix}\right.$ for a suitable ...
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1answer
24 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
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1answer
8 views

Cylindrical limits of integration for a particular triple integral

In cylindrical coordinates, what would be the limits of integration for the triple integral serving to find the volume of the region in $\mathbb R^3$ bounded by: $x^2 + y^2 = y$ and the sphere of ...
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2answers
37 views

Find all planes which are tangent to a surface

I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes ...
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2answers
26 views

Can every differentiable scalar function be written as a divergence of some vector field?

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field ...
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1answer
25 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
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0answers
6 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
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1answer
33 views

Interpreting limit notations

My question is: Are the following notations equivalent or not: $$(1)\;\;\;\;\;\;\text{When}\;||\textbf{x}||\rightarrow 0,\;\text{then}\;\;\;\frac{f(\textbf{x})}{||\textbf{x}||}\rightarrow0$$ ...
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1answer
12 views

$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$.

first of all i checked if (0,0) is critical point $Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $ now my idea was to replace $y$ with $xk$ because of the reduction of $F$ ,and find the hessian matrix to ...
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1answer
10 views

Proving that $2$-D parabolic coordinates are orthogonal

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. ...
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Setting up a double integral in terms of x and y to find flux

I am presented with the following problem, and it wants me to set up the double integral in terms of x and y, but I have no idea on how to continue solving this one, any ideas? Set up a double ...
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14 views

Can this multivariable function exist?

(3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles $x^2$ + $y^2$ = 4 and $x^2$ + $y^2$ = 10? If so, what is an example? If not, why not? I initially ...
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1answer
17 views

Independepnce of path in a closed curve line integral

Let $f(t)$ be a continuous function. Let $C$ be a smooth closed curve. Show that $$\oint\limits_C xf(x^2 + y^2)\,dx + y f(x^2 + y^2)\,dy = 0$$ Hint: Remember that $f(t)$ has a primitive function ...
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1answer
30 views

Proof of transformation law for double integrals

The second volume of Apostol's Calculus seems rather circumspect in its discussion of the change of variables formula for double integrals. Section 11.29 offers a proof under the following very ...
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4answers
190 views

Chain rule for partial derivatives intuition

Can somebody give me an intuitive explanation for the above equations. I'm not sure how they come about and how they can be perceived logically. $$\frac{\partial z}{\partial s} =\frac{\partial ...
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Partial derivative notation inside an integral - what would be normal?

What would be the most idiomatic way of writing the following idea? $$x^{t+u} = \int_0^x\int_0^{x}\frac{\partial (y^t)}{\partial y}\cdot \frac{\partial (z^u)}{\partial z}dz dy$$ for $\Re(x)>0$. ...
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Gaussian variance estimation via spectral decomposition

I was given a dataset (a mat file) of 100,000 observations, each with 50 dimensions (coordinates). Denote matrix $X$ is a 100,000x50 matrix in which each column was generated according to: ...
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Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
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15 views

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at ...
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2answers
26 views

Choosing path to show limit does not exist

I'm trying to show that the limit as $(x,y)$ go to $(0,0)$ for the function $f(x,y) = sin( x + y )/( |x| + |y|)$ does not exist. I initially tried the path $y=2$ and $y=1$, but I don't think I can use ...
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1answer
15 views

Finding flux across surface

Let S be surface {$(x,y,z) : x^{2} + y^{2} + 2z =2 . z \geq 0 $} Given F = $(y,xz,x^{2}+y^{2})$ n is outward normal .I have to find net flux through S . Since its closed surface so i applied Gauss ...
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24 views

About the $d\mathbf{s}$ notation

Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where ...
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Multivariable Optimization

I'm struggling to figure out how to go about this question: You are to produce a concrete box with an open top with a volume of 1$m^3$, having a wall and base thickness of $2$cm, by pouring concrete ...
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9 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
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0answers
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Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
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2answers
36 views

When does a double integral represent a surface area, and when does it represent a volume?

When does $\int_Af(x,y)dA$ represent a surface area geometrically, and when does it represent a volume? In my lecture notes I'm told it represent the volume underneath the surface $z=f(x,y)$, but I've ...
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3answers
41 views

What does it mean for a function to increase along a curve?

I think that if we were to say that, for instance, $y$ increases along the curve, (with no specific rate) then this means for the derivative to simply be positive. Or does it mean to choose the ...
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0answers
32 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
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Function with the opposite definition of Dirichlet function? [on hold]

I just happened onto the Dirichlet function today that states: $D(x)= \begin{cases} 0 & \text{if $x$ is irrational,}\\ 1 & \text{if $x$ is rational} \end{cases}$ which shows that points can ...
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Find Equation of plan passes through point and is perpendicular to the line

Find Equation of plan passes through point (5,2,1) and is perpendicular to the line x=4-t, y=1+2t, z=8-3t. I don't know if I did this correct but this is what I have: PQ = (4-5), (1-2), (8-1) = ...
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0answers
16 views

Calculating line integral by shortcut method

Here is the problem as I have been given it: A curve $C$ is given in Cartesian coordinates by $r(t)=(\cos(\sin(nt))\cos t,\; \cos(\sin(nt))\sin t,\;\sin(\sin(nt)))$, with $t$ between $0$ and $2π$ ...
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1answer
45 views

Is $f:\mathbb{R}^2\to\mathbb{R}$ differentiable on $(0,0)$?

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $f(x,y) = (x^2+y^2)\sin(\frac{1}{x^2+y^2})$ for $(x,y)\ne (0,0)$ and $f(x,y)=0$ for $(x,y)=(0,0)$. Is $f$ differentiable on $(0,0)$? So let's first ...
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17 views

Parametrize given curves

I'm given the following curves: $x = y^2 + 1$, $z = x + 5$ I'm eventually trying to find the unit tangent vector, so I need to find the r vector. Could I just assign $y = t$, and then have $<t^2 ...
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35 views

Probability distributions with closed-form cumulative distribution functions (CDFs)

I am interested in finding multivariate probability distributions for which the cumulative distribution functions (CDFs) are given in close form. For instance, the multivariate Gaussian distribution ...
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1answer
22 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
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1answer
11 views

continuity with 2-variables

The question is Determine whether $f$ can be defined at $(0,0)$ so that is is continuous $$f(x,y) = \frac{x^py^q + x^ry^s}{x^qy^p + x^sy^r}, p,q,r,s > 0$$. I chose numbers for p,q,r,s and ...
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1answer
30 views

Computing a specific line integral

Here is the problem as I have been given it: A curve $C$ is given in Cartesian coordinates by $r(t) = (cos(sin(nt))cost,\; cos(sin(nt))sint,\; sin(sin(nt)))$, with $t$ between $0$ and $2$$\pi$ ...
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0answers
30 views

Multi-variable function is undefined at every point, then limit still may exist

The following question was posed; If a multi-variable function $f(x,y)$ was undefined at every point on a curve, then could a limit exist of a point $(x_0, y_0)$ on this curve for this function? i.e ...
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Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
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7 views

Solve 1D wave equation on half-line using method of images

I'm trying to solve $\theta_t - D\theta_{xx} = f(x,t)$ on the half-line $0 < x < \infty$ for $0< t < \infty$ given boundary and initial conditions $\theta(0,t) = h(t)$, $\theta(x,0) = ...
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1answer
31 views

Partial differentiation or normal differentiation

Consider the function $$ f(x,y) = \begin{cases}\frac{xy(x^2-y^2)}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & \text{otherwise.}\end{cases} $$ Compute $$\frac{d^2f}{dxdy}(0,0)$$ and ...
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Using chain rule on a function $u:\mathbb R^n \rightarrow \mathbb R$

Suppose we have $x \in \mathbb R^n$, $\lambda \in \mathbb R$ and a function $u:\mathbb R^n \rightarrow \mathbb R$. I want to calculate the derivative $$ \frac{\partial u(\lambda x)}{\partial \lambda} ...
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2answers
39 views

$f(x,y):=\int_{a}^{xy}g(t)dt$ and $h(x,y):=\int_{y}^{x}g(t)dt$. Find $f'$ and $h'$

Suppose that $g:\mathbb{R}\to\mathbb{R}$ is continuous, $a\in\mathbb{R}$ is fixed and that $f(x,y):=\int_{a}^{xy}g(t)dt$ and $h(x,y):=\int_{y}^{x}g(t)dt$. Find $f'$ and $h'$. I don't know how I ...
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1answer
8 views

Gradient of the function $f(x)= \|x\|^p$

How can the gradient in this case be computed? I understand that $f(x)= (x_1^2 + x_2^2 + \ldots + x_N^2)^{p/2}$ but how do I proceed from here?
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Multivariable limits involving absolute values

Define $f$ on $\mathbb R^2$ by $$ f(x,y) = \begin{cases} f(x,y) = \frac{y|x|}x,& x\ne 0\\ 0,& x=0. \end{cases} $$ Is $f$ continuous a.) On the $x$-axis? b.) On the $y$-axis? c.) At ...
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1answer
40 views

Surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using parametrization

Calculate the surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using a suitable parametrization and the corresponding surface element. I think this set is a ...
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0answers
6 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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2answers
33 views

Stokes theorem and the simple closed curve on which work is maximum

I have a problem that states: Given the vector field $$\vec{F} = y^3\hat{i} + \left(4x - 2x^3 \right)\hat{j}$$ find the simple closed curve (with $\frac{d\vec{r}}{dt}\gt0$) on which the work ...
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1answer
31 views

Finding the local extrema of a function

One of our final exam exercise sheets features this particular exercise : Find the extrema of : $2xy^3+y-x^2=0$, where $y=y(x)$ . As I thought it, this exercise involves the implicit function ...