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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5 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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13 views

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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0answers
7 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
14 views

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
32 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
40 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
27 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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0answers
21 views

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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0answers
13 views

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
15 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
44 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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0answers
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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0answers
16 views

Derivative of a helicoidal trajectory

I am given the position vector: $r(t)=\frac{1}{3}\begin{pmatrix}R\space cos(\omega t)\\R\space sin(\omega t)\\v_0t\end{pmatrix}$ $R$=Radius $v_0$= veclocity in the z-direction I want to calculate: ...
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0answers
17 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$ ...
3
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0answers
28 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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0answers
14 views

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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0answers
6 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
30 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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0answers
15 views

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
37 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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0answers
30 views

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
39 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 ...
3
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1answer
27 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 ...
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1answer
25 views

Derivative of bilinear forms

I want to solve the following problems: Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$ ...
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1answer
31 views

Plane equation x units from point

I'm trying to find the equation of a plane normal to a certain vector $<x_1, y_1, z_1>$, and x units from a given point, $(a,b,c)$. Normally this question would be trivial, and I would simply ...
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1answer
27 views

Find diff. function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \forall n$. [on hold]

Give an example of a differentiable function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \ \forall n$. Here $R(x,y)$ denotes the error term while ...
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1answer
14 views

stokes theorm on intersection curve

Using stokes theorm, evaluate line integral $\int_L f.dr $ where L is intersection of $ x^2+y^2+z^2$=1 and x+y=0 traversed in counter clockwise direction when viewed from (1,1,0). f=yi+zj+xk. I ...
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0answers
74 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
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2answers
34 views

does anyone know how to graph $x^2+2y^2+3z^2=12$?

I just can't think of how I should draw this graph in 3 dimensions. Can anyone draw a graph for this?
2
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2answers
35 views

Limit of non-linear multi-variable function

I'm trying to prove the limit of the following function is $0$: $\lim_{(x,y) \to (1,-1)} {x^3} - {2xy^2} + 1$ I know that I'm trying to find a $\delta$ s.t $ 0 < \sqrt{(x - 1)^2 + (y + 1)^2} < ...
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0answers
33 views
+50

Is there a relationship between curl and area?

The cross product of two vectors is a new vector which lies on a new direction perpendicular to the plane of the multiplicand vectors. Its magnitude is the area of the parallelogram formed between ...
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16 views

Steepest descent from saddle point

I have the function $w(z)=\frac{1}{3}z^3+z$ where $z=x+iy$, i.e. a complex number. I am asked to find the saddle points of this function and then show the paths of steepest descent are ...
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1answer
36 views

Are these sets bounded or not?

Definition: A set $M \in R^n$ is bounded if there is a number C such that $|x| \leq C, \forall x \in M$. Problem: Determine if the following sets are bounded or not. 1) $\{ (x, y, z) : x^3 + y^3 + ...
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0answers
20 views

Velocity and acceleration problem

Show that if the dot product of the velocity and acceleration of a moving particle is positive (or negative), then the speed of the particle is increasing (or decreasing). If at all times t the ...
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1answer
27 views

Tangent plane and normals in $\mathbb{R}^2$

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function given by $$z = f(x, y) = x^4 + y^4.$$ Find the point on the surface $z = f(x, y)$, where the normal to the surface is perpendicular to the chord ...
2
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1answer
22 views

Inconclusive second derivative test at (0,0) for $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $

Second derivative test is inconclusive here , given f( x, y) is $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $ At (0,0) how do i check nature ? Also i would like to know general tactics when things like ...
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0answers
54 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
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0answers
48 views

Prove that exists $\delta>0$ such that, if $(x,y)\in S$ satisfies $\lVert(x,y) \rVert < \delta$, then $f(x,y) \leq f(0,0)$.

This exercise appeared on my Calculus II exam, and I didn't know even how to start doing it. Any hint is appreciated. Let $\ f, \ g : \mathbb{R^2}\to \mathbb{R}$ two $C^2$functions over the plane. ...
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1answer
26 views

Is every point of rational number boundary point?

While studying first chapter of multivariable calculus, I am wondering if every point of the rational number is boundary point. It is obvious that $\Bbb{R}^n$ is the union of interior, exterior, ...
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1answer
30 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
3
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1answer
50 views

Volume of a sphere with two cylindrical holes.

Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the ...
4
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2answers
44 views

Find that the limit is $0$

I have to prove that the following limit is $0$: \begin{equation} \lim_{(x,y)\to (0,0)}\frac{\lvert x\rvert^2y^2}{x^2+y^4}=0. \end{equation} This is a part of an exercise where I have to study the ...
1
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1answer
59 views

Integration: Step in paper unclear

I've seen in a paper the following step: $$2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx=(n-2)\int_{\mathbb R^n} |\nabla u|^2$$ This is not clear to me as I calculated: ...