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
77 views

Trying to understand Volume of a cone without the unit sphere

I have been working on the double integral proof for the volume of a cone. I found that I can use a unit-sphere Where the base of the cone is the equator and the height is the distance ${\rho}$ to ...
2
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2answers
111 views

Multivariate differentiability verification

I have tried an attempt on the following: Let $F:U\in \Bbb{R}^n\to \Bbb{R}$ and $f:\Bbb{R}\to \Bbb{R}$ where $f$ is an even function. Now $F(\mathbf{x})=f(|\mathbf{x}|)$, where $|\ . |$ is the ...
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1answer
678 views

Find a plane that passes through a point and is perpendicular to 2 planes

Find an equation of a plane that passes through $p(1,5,1)$ and is perpendicular to planes $2x+y-2z = 2$ and $x+3z=4$. I basically need the 2 other points to make the vector and perform the cross ...
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2answers
37 views

Find a parallel plane that contains a line

Find the equation of a plane that is parallel to the plane $5x-3y+2z=10$ and contains the line $x=t+4$ $y=3t-2$ $z=5-2t$ My attempt resulted in failure. Setting the directional vector: $v = ...
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1answer
25 views

Directional Differentiability follows from Multivariate Differentiability

The Exercise: "Suppose that a function $f: R^n \to R^m$ is differentiable at $x \in R^n$. Show that the directional derivative of $f$ in any direction $v\in R^n$ at $x$ exists and $D_vf(x)=Df(x)(v)$" ...
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1answer
42 views

Find the equation of a plane [duplicate]

Find the equation of a plane that passes through point $P(1,5,1)$, and is perpendicular to the planes $2x+y-2z=2$ and $x+3z=4$ My only guess so far is that we can obtain the plane's normal vector ...
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1answer
40 views

Proofs using vector properties

Let $a,b$ and $c$ be vectors in $\mathbb{R}^3$. How do I show that $$\|a-b\| \le \|a-c\|+\|c-b\|$$ and $$\|a \times b\|^2=\|a\|^2\|b\|^2-(a\cdot b)^2$$ ?
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1answer
30 views

Finding transformed region by change of variables

I have the equation of a curve $x^{2/3} + y^{2/3} = a^{2/3}$ and I'm using the change of variables $x = u\cos^3v $, $y = u\sin^3v$ . I have calculated the Jacobian ...
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1answer
44 views

Multivariable Calculus Vector Field Potential

How could I tell if the following vector field $$\vec{f}(x,y,z)=(xy+1)(e^xy e^yz e^z) \mathbf{i} \ + \ (x^2+xz)(e^xy e^yz e^z) \mathbf{j} \ + \ (xy+x)(e^xy e^yz e^z) \mathbf{k}$$ has potential without ...
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1answer
44 views

obtaining radius and distance from

A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the the x-axis. The curve traced out by ...
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2answers
44 views

Multivariable Calculus Length of Curve

I have to find the length of a curve C which is parametrized by $x(t)=\dfrac{e^t+e^{-t}}{2},$ $y(t)=\cos(t)$ and $z(t)=\sin(t)$ where $t$ goes from -1 to 5. I believe this involves simply finding the ...
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1answer
59 views

Extermum under constraint of parabula

Find the closet point on $2x^2-4xy+2y^2-x-y=0$ to the line $9x-7y+16=0$. Hint: the distance between $(x_0,y_0)$ to $ax+by+c=0$ is $d = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. For using lagtrange ...
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3answers
337 views

How to solve this integral for a hyperbolic bowl?

$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
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1answer
50 views

the space of exterior k-forms is infinite dimensional. why?

let Z be an n-dimensional smooth manifold with smooth (n−1)-dimensional boundary ∂Z, representing the space of spatial variables. Denote by $Ω^k$(Z), k = 0, 1, . . ., n, the space of exterior k-forms ...
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1answer
60 views

What does $\Delta$ mean in context of vector calculus?

I'm reading an article that has a formula for $\Delta \phi(x)$, where $\phi : \mathbf{R}^2 \rightarrow \mathbf{R}$ and $x \in \mathbf{R}^2$ and $\Delta \phi(x) : \mathbf{R}^2 \rightarrow \mathbf{R}$ ...
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2answers
6k views

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w ...
4
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1answer
223 views

What is the difference between vector-valued functions and parametric equations?

So as it is, I'm now starting to cover vector-valued functions in my Calculus III class. While studying the topic, I noticed that it seemed to be the exact same thing as parametric equations. I know ...
2
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1answer
100 views

Using Lagrange multipliers to maximize function

Use Lagrange multipliers to maximize function $$f(x,y)=6xy,$$ subject to the constraint $$2x+3y=24.$$ $$F(x,y,\lambda)=6xy+\lambda(2x+3y-24)$$ $$F_{x}=6y+2\lambda=0$$ $$F_{y}=6x+3\lambda=0$$ ...
2
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2answers
187 views

Find the limit of a multivariable function

The function is as follows: $f(x,y)=\frac{\ln(1+x^2y^2)}{x^2}$ and I want to calculate the following limit: $\lim_{(x,y)\to(0,y_0)}f(x,y)$ The reason I'm having trouble with this one is because ...
2
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1answer
89 views

Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
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3answers
417 views

If a sequences has two subsequenceswhich converge to to different limits then the sequence cannot be converging

I have already proved that if ${X_k}$ converges to a limit $L$, then any subsequence of it also converges to $L$. And now the question asks to show that if ${X_k}$ has two subsequence which converge ...
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1answer
275 views

Online classes/books in multivariable calculus?

So does anyone know of any good online courses in multivariable calculus? (Or in a possible alternative leap of curriculum, if said path has proven to be better/moar interesting.) I'm coming straight ...
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2answers
125 views

Probability distribution of dot product?

Sorry if this is a basic question. I don't know much about statistics and the closest thing I found involved unit vectors, a case I don't think is easily generalizable to this problem. I have a ...
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1answer
51 views

Did I solve this question about a line intersecting a plane correctly?

I'm asked to find if there is any point of intersection, and if so, where it is between the line represented by the symmetric equation $\frac{x-3}{3}=\frac{y+1}{-2}=\frac{z-10}{4}$ and the plane ...
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1answer
42 views

Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
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1answer
54 views

Find a limes - function with 2 variables

1)$ \displaystyle \lim_{x \to 1 y \to 1} (2y-x)^{\frac{1}{\sin(2y-x-1)}}$ 2)$ \displaystyle \lim_{x \to 0 y \to 1} (2x+y)^{\operatorname{ctg}(2x+y-1)}$ Ok, how to find a limes or prove that it ...
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3answers
86 views

Inequality for Integral of Vector Function

I am trying to prove $\| \int^{b}_{a} \vec{r}(t) dt\| \leq \int^{b}_{a} \| \vec{r}(t) \|dt$. I am fairly certain that this can be derived from the Cauchy-Schwarz inequality, but I can't quite ...
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0answers
50 views

Derivative in non orthogonal coordinates

I am trying to transform irregular shape in common Cartesian coordinates ($x-y$) into a regular shape in a generalized coordinates(e.g.,$u-v$), in which the transform can be defined as $u=u(x,y)$ and ...
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1answer
66 views

Applying the implicit function theorem

Let $F:\mathbb{R}^2\to\mathbb{R}$ of class $C^2$ and $\displaystyle\frac{\partial f}{\partial v}(u,v)\neq0\; \forall(u,v)\in\mathbb{R}^2$. If $(x_0,y_0,z_0)\in\mathbb{R}^3$ is such that ...
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2answers
65 views

Area with double integral.

$f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=(x^2+y^2)^2-8(x^2-y^2).$ Find the critical points of $f$ and the area of $X=\{(x,y):f(x,y)\leq 0\}$. To find the critical points I just have to find ...
2
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1answer
27 views

Find the differential of an n-variable function

The problem goes like this: If $f:\mathbb{R}^n\to\mathbb{R}, f(x)=\arctan||x||^4$, prove that $Df(x)(x)=\displaystyle\frac{4||x||^4}{1+||x||^8}$ Now, I've calculated each of the partial ...
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1answer
36 views

Stokes's Theorem on a Curve of Intersection

Let $C$ be the intersection of $y+z=0$ and $x^2+y^2=a$ ($a>0$), oriented counterclockwise when viewed from above on the $z$-axis. Compute $$\int_C(xz+1)\text{ d}x+(yz+2x)\text{ d}y$$
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2answers
40 views

Find the limit of a multivariable function that's a quotient of polynomials

The functions is as follows: $f(x)=\frac{x^3y^3}{x^2+y^2}$ Now, the limit at $(0,0)$ does exist and it's $0$ but I only know that from WolframAlpha. I've tried all fractional manipulations I can ...
4
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2answers
63 views

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...
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2answers
584 views

Problems with limits of functions of two variables

I have the following function: $$f(x,y):=\begin{cases}\frac{x^3y}{x^6+y^2}&,\;\;(x,y)\neq (0,0)\\{}\\0&,\;\;(x,y)=(0,0)\end{cases}$$ I'm asked about continuity at the origin and the limit of ...
3
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1answer
185 views

Jacobian of a Composition involving a Linear Transformation

Let $f:{\mathbb{R}^n} \to \mathbb{R}$. For each $z \in {\mathbb{R}^n}$ define $\tilde f\left( z \right) = f\left( x \right)$, where $x = Az + s$, for some $A \in {\mathbb{R}^{n \times n}}$, $s \in ...
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1answer
42 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
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0answers
24 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
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0answers
115 views

Tricky calculus exercise (Now finding a tangent space)

Let $f(x,y,z)=z^3-g(x,y)z+h(x,y)$, with $g,h:\mathbb{R}^2\to \mathbb{R}$ of class $C^1$ and $g(x,y)>0\; \forall (x,y)\in\mathbb{R}^2$. Consider $S=\{(a,b,c)\in\mathbb{R}^3:f(a,b,c)=0\}$, ...
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2answers
64 views

How plot $f(x,y) = \frac{x}{1-y}$ with $x^2+y^2<1$?

How do you plot $$f(x,y) = \frac{x}{1-y} \text{with}~ x^2+y^2<1$$ in Mathematica or Maple?
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1answer
126 views

Is $\cos\sqrt{xy}$ uniformly continuous?

I'm trying to find out if $$ f(x,y)=\cos\left(\sqrt{xy}\right) $$ is uniformly continuous on the set $\{(x, y)\in\mathbb{R}^2 : x\geq0, y\geq 0\}$. The theorems I have available to use for this are ...
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1answer
76 views

Multivariable Calculus Order of Integration Question

I have a triple integral $ \int_{0}^{2} \int_{0}^{y^3} \int_{0}^{y^2} f(x,y,z) \ dz \ dx \ dy $. I have to find five different iterated integrals equivalent to this integral. I know that order of ...
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2answers
86 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
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0answers
22 views

Multivariable Calculus Order of Integration [duplicate]

I posted a question yesterday which was! Multivariable Calculus Order of Integration Question After comments, the solutions I came up with were: \begin{align} \int_{0}^{2} \int_{0}^{y^2} ...
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2answers
82 views

Could a computer theoretically compute all integrals in terms of some special functions or it is not possible theoretically?

Could a computer theoretically compute all integrals in terms of some special functions or there need to be exist infinite number of such special functions to represent all integrals? I know there ...
2
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1answer
31 views

Multivariable limit of $\frac {xy}{e^{x^2y^2}}$

We want to find $$ \large \lim_{x^2 + y^2 \to \infty } \frac {xy}{ e^{x^2y^2} }$$ It looks like it goes to zero but if we let $y = \frac 1x$then the limit is equal to $\frac 1e$ i.e. the function ...
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0answers
79 views

Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the Reynolds or Lubrication Equation, but ...
7
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1answer
86 views

Two-variable limit, quotient of polynomials

I'm trying to evaluate the following limit, $$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$ which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ ...
2
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1answer
169 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
2
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1answer
35 views

Divergence as a surface integral

I had a shot at the final problem of section 16.4 in 'Calculus a Complete Course' by Adams, I knew full well that I wasn't even going to come close to a correct answer so my question relates to the ...