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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1answer
18 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? ...
0
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1answer
9 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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0answers
12 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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1answer
27 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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1answer
32 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 ...
3
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0answers
20 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
4 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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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
28 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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0answers
27 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
6 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?
1
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1answer
57 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: ...
8
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1answer
1k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
1
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2answers
23 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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0answers
4 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 ...
0
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1answer
358 views

The motion of a solid object

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density at the point and occupies a region W, then the ...
3
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1answer
25 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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2answers
74 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
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1answer
24 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
18 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 ...
1
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1answer
268 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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1answer
26 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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0answers
26 views

Lagrange multiplier over two constraints

I'm having two constraints $g_{1}$=$x+y-z+2=0$ and $g_{2}$=$z^{2}-x^{2}-y^{2}=0$ and I want to determine the point on the intersection which is closest to the origin. The question asks us to use ...
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1answer
13 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 ...
2
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2answers
805 views

Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
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0answers
70 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
83 views

Find the absolute extrema by Lagrange multipliers

Find the absolute extrema for the function $g(x,y)=e^{x^2}-y^2$ on the unit disk $D$ given by: $D=\{(x,y)|x^2+y^2\le1\}$. Do this by first finding all critical points of and classifying them, then ...
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2answers
33 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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2answers
33 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?
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1answer
1k views

Find the orthogonal projection of b onto a

Given $${\mathbf a} = (-10, -8, 9) $$and $${\mathbf b} = (4, 5, 8) $$ Can someone please define what it means to find the orthogonal projection of b onto a? Also what is the formula for computing ...
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0answers
24 views

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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0answers
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 + ...
3
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1answer
49 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 ...
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0answers
19 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 ...
2
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1answer
21 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 ...
0
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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 ...
0
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1answer
54 views

Differentiating a matrix function with respect to a scalar

I would like to differentiate the following with respect to psi (partial): $$ \operatorname{trace}\bigl((X^\top X)^{-\psi} P\bigr). $$ Here we have that: $ X \in \mathbb{R}^{p \times n}, P \in ...
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1answer
26 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
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0answers
52 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
43 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
28 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 ...
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3answers
42 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
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1answer
26 views

An unusual “multivariate Gaussian integral” that comes up when trying to translate results about a standard Gaussian to the general case

I am trying to solve this question and it leads me to a strange looking integral that I do not know how to solve. Let $\Sigma$ be positive semidefinite, and $1>\lambda>0$. I am not certain I am ...
2
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3answers
55 views

How to draw a contour map of $ f(x,y)=x^2+y^2+xy$

I have used a program to see that it is an ellipse but I want to know the process of thinking to actually draw the contour map myself. $x^2+y^2+xy=C$ for $C=0,1,2,3,...$ I can't seem to get it into ...
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2answers
43 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 ...
2
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1answer
29 views

If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$

Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$. If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The ...
1
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1answer
19 views

Trick to finding length of parametric curve

I was giving the parameters of the curve: $x = 2cos(2t)$ $y = 2sin(2t)$ and $z = 1$, where $ 0 \leq t \leq 10 \pi$ This curve describes a cylinder in the $z$ direction, and seems very straight ...