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
29 views

Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
1
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0answers
12 views

How to approach this problem by directly using line integral and the Green's Theorem?

The question No 4 is asking to use methods: one just directly solving by line integral and another is by using Green's Theorem. Thanks in advance!
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0answers
27 views

How many max/min/saddle points of $f(x, y)$ on a region?

Consider the function $f(x,y)=2xe^x\sin y$ on the region $\{(x,y) \mid -\frac{\pi}{4} \leq y \leq \frac{3\pi}{4}\}$. How many maximums/minimums/saddle points are there? I honestly have no idea how ...
1
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0answers
32 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
2
votes
2answers
42 views

Two and Three Variable Limit Questions

Find the following limits, if they exist. $$\lim_{x,y\rightarrow 0,0}\frac{x^2 + \sin^2 y}{\sqrt{x^2+y^2}}$$ I believe we're suppose to use the squeeze theorem on this first one above. Possibly ...
0
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1answer
29 views

Multivariable-calculus, derivative and second derivative [closed]

I got the function $f(x,y)=\ln \sqrt{x^2+y^2}$. The task is to find the derivative function and the second derivative function. How do I get there?
3
votes
2answers
40 views

Example of a function $F(x,y)$

I'm trying to find a non trivial function $F(x,y)$ such that $div F(x,y)=0$ everywhere and $F(x,y)=0$ on the unit square. I know that there are some books that provide such example but I didn't find ...
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2answers
43 views

Multivariable-calculus

The task is the attached image. We got the function and the domain of definition. The task is to decide the function´s lowest value and biggest value plus the range. Lowest value should be -sqrt3,6 ...
1
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1answer
26 views

Surface integral on sphere

Is there a direct way to calculate the surface integral of the gradient of some smooth function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ over the sphere $S^2$ without knowing $f$? $$ ...
0
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2answers
36 views

Simple multivariable limit problem

How do I evaluate the following limit? $$\lim_{(x,y)\to(0,0)} \frac{x^3}{\sqrt{x^2+y^2} (x^2+y^2)}$$ It seems like this limit would be some finite nonzero number, but I don't have too much experience ...
0
votes
2answers
20 views

Marginal density function question

The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, ...
0
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0answers
21 views

Find the extreme value of $f(x,y,z)=x(y+z)$ on the curve of intersection of the right circular cylinder $x^2+y^2=1$ and the hyperbolic cylinder $xz=1$

Using the concept of Lagrange multipliers, we can treat the circular cylinder as $G$ and assign it $\lambda$, and treat the hyperbolic cylinder as $H$ and assign it $\mu$ Then $\Delta$f=$\lambda G + ...
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2answers
30 views

Find the centroid of the boomarang shaped region for the parabolas $y^2=-4(x-1)$ and $y^2=-2(x-2)$

I know the formulas, I only need assistance setting up the initial integral. So my order of integration must be $\mathrm{d}x$ $\mathrm{d}y$. Then if we solve the parabola for $x$ the new integral we ...
3
votes
0answers
19 views

Converting a polar integral to spherical

$$\int_0^{2\pi} \int_0^{\sqrt{2}}\int_r^{\sqrt{4-r^2}}\mathrm{d}z \, r \, \mathrm{d}r \, \mathrm{d}\theta$$ So in spherical this would become: $$\int_0^{2\pi} \int_0^{\pi/4}\int_0^2 \rho^2\sin\phi \, ...
0
votes
1answer
45 views

Use the transformation $x=u^2-v^2$, $y=2uv$ to evaluate the integral

$$\int_0^1 \int_0^{2\sqrt{1-x}} \! \sqrt{x^2+y^2} \, \mathrm{d}y\,\mathrm{d}x$$ Here's where I'm at: $J(x,y)=4u^2+4v^2$ Substituting $x$ and $y$ into the integral: $\sqrt{(u^2-v^2)^2+4u^2v^2} ...
0
votes
1answer
66 views

Splitting Integral into Two Parts

This question might seem very simple, but I can't seem to figure it out. Suppose I have an integral over a square region. I was wondering in which case it would be incorrect to split the integral into ...
8
votes
2answers
113 views

Multivariable limit with logarithm

I have to prove that the limit $$\lim_ {{(x,y)} \to {(0,0)}} \frac{xy^2\ln\frac{|x|}{|y|}}{{(x^2+y^2)}^{\frac 12}}$$ does not exist. I've tried to find two different paths that show that the limit ...
1
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2answers
39 views

Multivariable-calculus. Find the stationary point (critical point)

I got the function $f(x,y)=\ln(1+x^2+y^2)$. The task is to find the stationary points. The correct answer is $(0,0)$. How do I find the stationary point? I`ve differentiated it at put it equal to zero ...
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0answers
87 views

Proof of Clairaut's Theorem (Weaker Statement)

I would like to ask if such a proof of Clairaut's Theorem (Weaker Statement) is correct Statement: Suppose $f$ is a real-valued function of two variables $x,y$ and $f(x,y)$ is defined on an open ...
0
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2answers
55 views

Why are the partial derivatives of $f(x,y)=xy/(x^4+y^4)$ equal to $0$ at $(0,0)$?

Read "$x=0$ or $y=0 $" as $x=0\cup y=0$ I don't understand the section of the solution highlighted in green. For instance if: $f(x)=e^{-1/x^2}$ for $x\neq0$ and $f(0)=0 $ I cannot just say ...
0
votes
1answer
29 views

How to find the surface element for the cylinder $x^2 + y^2 = r^2$?

So if given a surface (cylindrical) which has radius r and equation $x^2 + y^2 = r^2$, I want to work out the line element for it. How do I get it? I know the final answer has to be $dS^2 = r^2dϕ^2 ...
0
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0answers
30 views

Calculating area using change of variables

Let $X(x,y) = (1,1)$ and let $(u,v) = F(x,y) = (x-y^2,2y)$. Find $F_*X$ in terms of u and v. Find $F^*du\land dv$. Use the change of variables $(u,v) = F(x,y) = (x-y^2,2y)$ to calculate the area in ...
0
votes
1answer
25 views

Vectorial Calculus proof [closed]

Please help me to prove the following identity wherein $\phi$ is a scalar field and d$\vec l$ is the linear element: $$\int \nabla \phi \cdot d\vec l = \int d\phi$$ hopefully step by step. ...
0
votes
1answer
20 views

Maximising a function under a constraint

Let $$f(x,y,z) = 4x+2y+5z^2 \text{ and } A=\{(x,y,z) \in \mathbb{R^3} ; \, x^2+y^2+z^4 \leq 5 \}.$$ Find the maximum of $f$ on $A$. My question is the following: How do I prove that the maximum must ...
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5answers
52 views

Vector field ${\bf F}$ with $\int_S {\bf F}\cdot{\bf n}\ dS=c$

Find a vector field ${\bf F}$ on $ {\bf R}^3$ with $$\int_S {\bf F}\cdot{\bf n}\ dS=c > 0 \tag{1} $$ where $S$ is any closed surface containing $0$ and ${\bf n}$ is normal Here there is a ...
0
votes
1answer
35 views

Deriving FTC from the generalized Stokes.

How do I derive the Fundamental Theorem of Calculus from the generalized Stokes theorem?
0
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1answer
67 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
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2answers
45 views

Lagrange multipliers/multivariable optimisation problem

Problem: Maximise the volume $V$ of a cuboid shaped box with closed top, fixed surface area $S$, and side lengths $x, y,$ and $z$ What I've got so far: $V=xyz$, $S=2(xy+yz+zx)$, $\nabla ...
5
votes
1answer
135 views

Exchange the order of the two limits

Suppose both limits exist, when is it true that $$\lim_{x\to a}\lim_{y\to b} f(x,y) = \lim_{y\to b}\lim_{x\to a} f(x,y) ?$$ and further when is it true that these two limits are equal to ...
0
votes
0answers
20 views

Second partial derivative of the 2-norm of a multivariate function [on hold]

I want to calculate second order partial derivative of a function $g(x,y)=\|f(x,y)\|^2$ with respect to $x$ and $y$ in case of $g$ and $f$ are either vector or matrix. Is it possible? Thank you in ...
2
votes
1answer
43 views

Finding the partial derivatives of this function

Let $g \in C^1(\mathbb R)$ be a real valued function and $f$ defined by $$f(u,v,w) = \int_{u}^{v} g(w^2+\sqrt{s})\,\,ds$$ where $u,v,w \in \mathbb R$ and $u,v>0$. Find all partial derivatives. I'm ...
1
vote
1answer
23 views

Convexity on a direction

In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have: A function $f$ defined on a triangle $T$ is said to be convex in the ...
1
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0answers
23 views

Maximum and minimum of function of two variables on the set $M=\left\{(x,y)\in \mathbb R^2 \ \Bigl|\ x+y\leq L\right\}$

Find the points of maximum and absolute minimum of the following function of two variables $$f(x,y)=\frac{\cosh (x-y)}{x}\left(e^{x}-x-1\right)$$ on the set: $$M=\left\{(x,y)\in \mathbb R^2 \ \Bigl|\ ...
2
votes
1answer
69 views

Definitions of differentiability

I have seen two definitions of differentiability of a real valued function and I wonder why they are equivalent. The first definition: For a function $\mathbf{F}:\mathbb R^n\to \mathbb R$ is ...
0
votes
2answers
31 views

Jacobian and uniform distribution

If I perform a change of variables for independent and uniform random variables $x, y, z$, which is something like the following: $$u=x-y+z/2 \\ v=z/2 - x \\ w = z/2$$ then how do I know whether ...
0
votes
1answer
35 views

Finding a closed line integral using Stokes' Theorem

Find the line integral $\int_C \vec{F} \cdot \vec{dS}$, where $C$ is the circle of radius 3 in the $xz$-plane oriented counter-clockwise when looking from the points $(0, 1, 0)$ into the plane and ...
-1
votes
2answers
45 views

Evaluate the flux integral [closed]

Evaluate the flux integral $$ \int\!\!\int_{S} {\rm curl\left(\vec{F}\right)} \cdot \vec{dS} $$ where $$ \vec{\rm F}(x, y, z) =\langle xe^{y^2}z^3 + 2xyze^{x^2 + z}, x + z^2e^{x^2 + z}, ye^{x^2+z} + ...
0
votes
0answers
35 views

What is the density of a homogeneous disk with mass $m$ and radius $a$?

Could someone help me understand why the density of a homogeneous disk is $\dfrac{m}{(\pi a)^2}$? I am trying to understand an example about finding the moment of inertia of an object. The question ...
2
votes
2answers
42 views

Direction of Greatest Increase

Problem: Find the direction of greatest increase at $P$. $$f(x,y)=4x^2+y^2+2y$$ $$P=(1,2,12)$$ Solution: The greatest increase in $f(x,y)$ at $P$ can be attained by moving in the direction of ...
-1
votes
0answers
19 views

Transformations of variables

I don't know how apply the second transformation of variables in this integral. The integrated function can be assumed to be $N_{MJ}(\bar\xi)*D^{-1}(\bar\xi)*\bar\xi_i*\bar\xi_n=\bar\xi_1* \bar\xi ...
1
vote
1answer
24 views

A strictly convex function defines an implicit function with non-positive second derivative

Problem. Let $F\colon \mathbb{R}^2 \to \mathbb R$ be a non-negative, $C^2$ function which is also strictly convex, meaning that $$ F(\lambda P + (1-\lambda)Q) < \lambda F(P) + ...
-1
votes
1answer
37 views

Prove that $f(x,y) = 1/(x^2 + y^2)$ has limit $\infty$at $(0,0)$

The question is: "prove that function $f(x,y) = $ $1 \over (x^2 + y^2)$ have limit $\infty$ in point $[0,0]$. This is pretty standard question, and even my book answers it immediately afterwards, ...
0
votes
2answers
47 views

A limit two variables

How can I prove that $\displaystyle\lim_{(x,y)\to(0,0)}\dfrac{\mathrm{e}^{(a+x)(b+y)}-\mathrm{e}^{ab}-b\mathrm{e}^{ab}x-a\mathrm{e}^{ab}y}{\sqrt{x^2+y^2}}=0$
-1
votes
1answer
35 views

Find finite area between curves [closed]

Find the finite area enclosed between $r= a \sin 4(\theta)$ and $r= a \sin 2(\theta)$ in polar coordinate system.
2
votes
1answer
15 views

Generalizing the total differential to multidimensional codomains

Consider a function $f: \mathbb{R}^n \to \mathbb{R}$ in the variables $x_1, \, x_2, \, \dots, \, x_n$. In multivariable calculus, we learn that the total differential of $f$ is defined as $$ df = ...
2
votes
1answer
39 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
0
votes
1answer
25 views

If $f(x,y,z)$ takes a maximum $f(x_0,y_0,z_0)$ under $g(x,y,z)=1$ then $\nabla f$ is parallel to $\nabla g$ at the point under any condition?

One of my exam questions says If $f(x,y,z)$ takes a maximum $f(x_0,y_0,z_0)$ under $g(x,y,z)=1$ then prove $\nabla f$ is parallel to $\nabla g$ at the point. Is it true general? I think on surface ...
0
votes
0answers
41 views

How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
0
votes
2answers
24 views

Finding a value R that maximizes the flux a vector field over half a sphere of radius R

Sorry for the bad title, couldn't think of a less convoluted way of writing it. I have to find $ R\in \mathbb{R}$ so that the flux of $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
0
votes
1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...