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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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
votes
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 ...
1
vote
3answers
414 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 ...
4
votes
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 ...
4
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
3answers
85 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 ...
0
votes
0answers
47 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 ...
1
vote
1answer
64 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 ...
2
votes
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
votes
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 ...
0
votes
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 $$\int_C(xz+1)\text{ d}x+(yz+2x)\text{ d}y$$
0
votes
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
votes
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 ...
3
votes
2answers
582 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
votes
1answer
184 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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 $S=\{(a,b,c)\in\mathbb{R}^3:f(a,b,c)=0\}$, ...
0
votes
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?
1
vote
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 ...
1
vote
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 ...
0
votes
2answers
85 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 ...
0
votes
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} ...
0
votes
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
votes
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 ...
4
votes
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
votes
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
votes
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
votes
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 ...
7
votes
1answer
395 views

Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
1
vote
1answer
36 views

How can I show that $ \int_{\Gamma_1}F\cdot dr-\int_{\Gamma_2}F\cdot dr=2k\pi $?

Let $F:{\Bbb R}^2\to {\Bbb R}^2$ be such that $$ F(x,y)=\left(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}\right). $$ Suppose we have to one-to-one $C^1$ curves: $\gamma_j:[0,1]\to{\Bbb R}^2$, such ...
0
votes
1answer
43 views

Implicit Function!

I need to show that equation $z^{3} + z + xy=1$ defines an unique function on the set of real numbers $g(x,y)=z$ ,for any x,y.Also i need to find $g'(1,1)$.This is what i have so far: ...
0
votes
2answers
82 views

Surface of graph of a function.

Let $G:=\{(x,y,z)\in\mathbb{R}^3:|x|<|z|^2,|y|<|z|,0<z<1\}$ and $f:G\to\mathbb{R},f(x,y,z)=2x+2y+z^3$. Calculate the surface of the graph of f. We recently got introduced to Stokes' ...
2
votes
0answers
56 views

smooth curves in $\mathbb{R}^2$

how do I show that $x^2-y^2= 0$ is NOT smooth in any neighbourhood of the the point $(0,0)$? I just thought of taking the gradient and showing it is zero at the origin. But I'm not sure if that's the ...
1
vote
1answer
64 views

Extrema with Lagrange Multipliers

I have the following exercise: Define $f:\mathbb{R}^2\to\mathbb{R}$ by $f(x,y)=(x^2y)^{1/3}$. Is $f$ differentiable at $(0,0)?$. Has $f$ absolute extrema in $D=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq ...
0
votes
1answer
69 views

check if complex function is differentiable

The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the ...
1
vote
1answer
59 views

Integrating function over half-ball

Let $B:=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2\le 4, z\ge 0\}$. Calculate $\int_B \dfrac{(x^2+y^2)z}{(x^2+y^2+z^2)^2}d\lambda_3(x,y,z)$. I see that the boundary of $B$ is ...
0
votes
0answers
21 views

Partial derivatives involving unrelated variables

I have that y(x, t) := F (x − ct) + G(x + ct), and I need to find ∂2y/∂t2 and ∂2y/∂x2. I'm a bit puzzled, because 't' isn't held constant when calculating ∂y/∂x and vice versa. My guess is ∂y/∂x = ...
0
votes
0answers
31 views

what is the result of this Integral with polar coordinates?

I can't understand where am I going wrong with this integral. The final answer should be 4 but I get 2/3. Am I wrong or is the teacher wrong? Given this function: $f(x,y)=x+y$ and this domain D ...
2
votes
1answer
33 views

Polar Integral Confusion

Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer. Question Use a polar integral to find the area inside the ...
1
vote
1answer
41 views

How can I calculate the maximum and minimum of this function?

This is incorrect and I have a feeling I am not doing this correctly.
1
vote
1answer
27 views

Differentiating the Spherical Mean w.r.t its radius

this question regards differentiating the spherical mean with respect to its radius. This is my attempt so far: Start with the equivalent form of the spherical mean so that we can pass the partial ...
1
vote
2answers
66 views

Multivariable Calculus - Calculating Derivative Matrix

I'm working with Munkres' Analysis on Manifolds. From chapter 2 (this isn't a homework question): Given $f: \mathbb R^2 \rightarrow \mathbb R^2 : f(r,\theta)=(r\cos(\theta),r\sin(\theta))$, ...
2
votes
2answers
418 views

Local versus global implicit function

Suppose the equation $f\left(x,y\right)=0$, with $x\in I_{1}$ and $y\in I_{2}$, $I_{1}$ and $I_{2}$ being open intervals. Additionally, consider that the conditions required to apply the Implicit ...
1
vote
1answer
19 views

convert equations from 1+j+k format and finding area

The specific problem is: find the area of a triangle having the vertices $C(1,0,1)$, $B(0,2,3)$, and $A(-1,5,-2)$. I've tried this so far: obtained the vectors $BA (-1,3,-5)$ and $BC (1,-2,-2)$ The ...
1
vote
2answers
74 views

how to solve $\frac{df}{dx}$ derivative against $dy$?

If I take derivative of $\frac{df}{dx}$ against ${dy}$, will it be $\frac{d^2f}{dxdy}$. I am a little confused. So to solve this we will look for those values which have variable $y$ in them in ...
5
votes
3answers
5k views

How do you graph $x + y + z = 1$ without using graphing devices?

How can I graph $x + y + z = 1$ without using graphing devices? I equal $z = 0$ to find the graph on the xy plane. So I got a line, $y = 1-x$ But when I equal 0 for either the $x$ or the $y,$ I get ...