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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3answers
32 views

$\lim _{(x,y)\to(0,0)}\frac{x(x-y)}{x^4+y^4}$ does not exists

Show that $$\lim_{(x,y)\to(0,0)} \frac{xy(x-y)}{x^4+y^4}$$ does not exists. I've tried the "traditional" paths, with $(x,x)$, $(0,x)$, $(0,-x)$, but I only get $0$ as answer. Any hint? Thanks!
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1answer
25 views

$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$be continuous and define$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by $g(x)=|f(x)|$ prove g is continuous

This was assigned as a practice problem for my multivariable calculus class, and its really not making sense to me, can someone help me out Let $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a ...
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2answers
36 views

How do I simplify this expression $\exp(xy)-1=\ln\left((x^2+y^2)^{1/2}\right)$

How do I simplify this expression $\exp(xy)-1=\ln\left((x^2+y^2)^{1/2}\right)$. I try some ways but could simplify, can you try help me? Edit: by simplify I mean put y on one side and x on another ...
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1answer
19 views

Chain rule for partial derrivatives

Suppose $$t' = t$$ $$x' = x - vt$$ I need to prove that $$\frac{\partial{}}{\partial{x}} = \frac{\partial{}}{\partial{x'}}$$ $$\frac{\partial{}}{\partial{t}} = \frac{\partial{}}{\partial{t'}} - v ...
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0answers
8 views

Finding a limit using Taylor's theorem

let's say that g(x,y) is $c^{n+1}$ and let's say that p(x,y) is it's n-th order Taylor polynomial. I am trying to prove that: $$\lim_{(x,y)\to (0,0)} \frac{g(x,y)-p(x,y)}{(\sqrt{x^2+y^2})^n}=0$$ I ...
3
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2answers
23 views

What is the domain of this multivariable function?

Let $h(x,y,z) = (z^2 -xz + zy -xy)^{1/4}$. What is the domain on this function? I know that \begin{align*} z^2 -xz + zy -xy \geq 0 \\ \implies z(x+y) -x(z+y) \geq 0 \\ \implies (z-x)(z+y) \geq 0 ...
0
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1answer
21 views

Using $\epsilon - \delta$ method prove that $\lim_{(x, y)\to (\frac{\pi}{2}, 0)}xy\sin(x + y) = 0$

Using $\epsilon - \delta$ method prove that, $$\lim_{(x, y)\to (\frac{\pi}{2}, 0)}xy\sin(x + y) = 0$$ Here's what I tried: $$|xy\sin(x + y)| \le |xy| \le \frac{|x|^2 + |y|^2}{2}$$ But I can't ...
1
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2answers
32 views

Double integral of $y / (x^2 y^2 + 1) dx dy$

I'm trying to solve the double integral $\displaystyle \int_0^1 \int_0^1 \frac{y}{x^2 y^2 + 1} dx \, dy$. I'm guessing something with natural log will have to be done. Doing the steps of this problem ...
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0answers
25 views

Partial Derivative Chain Rule verify solution

for a function $g(u,v) = f(x(u,v),y(u,v))$ find $\frac{\partial^{2} g}{\partial u \, \partial v}$ To begin with, we know by the chain rule that the first order partial derivative is: $$ \frac{ ...
1
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0answers
36 views

How to solve limits?

The above limit was solved by making a seemingly arbitrary substitution. The previous limit was solved by making a linear substitution $y=mx$. Which again seemed a bit out of the blue. For another ...
2
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2answers
29 views

Notation on partial deriviatives

If I need to find $$\frac{\partial ^{2}g}{\partial u \partial v}$$ Then do I want to perform $$ \frac{\partial} { \partial v}\ \big( \frac{\partial g}{\partial u} \big) $$ or $$ \frac{\partial g} ...
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1answer
23 views

What's the domain of this function?

$$f(x,y,z) = \frac{1}{x-z}$$ I see two possible domains: $$D = \{x,y,z \in \mathbb{R}\mid x \neq z\}$$ or $$D = \{x,z \in \mathbb{R}\mid x \neq z\}$$ Is it equally valid to say the function has ...
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0answers
27 views

Quick Question: dot product with del

Is $(v \cdot \nabla)F = (\nabla F) \cdot v$? I'm not quite sure.
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1answer
8 views

Contour plot of a parabolic cylinder?

Consider the following function: My book says this is its contour plot: I don't understand why. I would have expected the lines to get closer together as you get closer to the X-Axis, yet they ...
0
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1answer
21 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
2
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1answer
38 views

Doubts on Differentiation in $\mathbb{R}^p$

I am currently reading R.G Bartle's "Elements of Real Analysis" for a one semester course in Advanced Real Analysis. In the chapter on Differentiation in $\mathbb{R}^p$, I am confused regarding the ...
-3
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0answers
29 views

Help with stokes theorem question [on hold]

The plane $z=x+4$ and the cylinder $x^2+y^2=4$ intersect in a curve $C$. Suppose $C$ is oriented counterclockwise when viewed from above. Let: ...
-1
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2answers
62 views

How to prove a limit exists using delta and epsilon?

I am having issues writing a proof using deltas and epsilons for this limit: I basically do not how to construct a proof after finding a limit. My main problem with problems like that arise from the ...
2
votes
3answers
67 views

The projection of a vector value function onto the xz-plane.

Okay, so I missed CalcIII today and I'm struggling a bit here. $r(t) = (\sin t,\cos t,7\sin t + 4\cos 2t)$ Find the projection of $r(t)$ onto the xz-plane for $−1 \leq x \leq 1$ Answer as an ...
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0answers
28 views

A function of class $C^2$

Problem: given a function $F:\mathbb{R}^2\mapsto\mathbb{R}$ of class $C^2$, with $F(0,0)=0,\nabla F=(2,3)$, shown that a surface $F(x+2y+3z-1,x^3+y^2-z^2)=0$ can be given locally at $(-2,3,-1)$ as ...
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1answer
15 views

Find parametric equations using parallel lines and line through a point

How would I find the parametric equation of a line through $(1,-1,1)$ and parallel to the line $x + 2 = 1/2y = z -3$. Would I find the vector equation first? If so, how would I go about doing that?
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0answers
18 views

Mobius band parameterizaton: Showing injective

So, I'm trying to show that the parameteization function from $\mathbb R^2$ to $\mathbb R^3$ given in the wikipedia page http://en.wikipedia.org/wiki/Mobius_band#Geometry_and_topology is injective on ...
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2answers
32 views
+50

Proof of Gauss theorem (divergence theorem) in $\mathbb R^2$

I am trying to solve an exercise in where it is asked to show the divergence theorem, or also known as Gauss theorem, in $\mathbb R^2$ using Green's theorem. I suppose that the divergence theorem in ...
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0answers
8 views

Negative Gaussian Log Likelihood Minimization - question about scaling and Hessians

I received this question from my niece, and had to admit it's quite over my head. I wonder if anyone here could assist? I am doing a minimisation with the negative of a gaussian log likelihood ...
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0answers
16 views

locus of all points on the surface of a cone

I need some help to find the locus of all points on the surface of a cone with semi angle theta and with its axis along the direction given by unit vector v. I need to use the definition of the dot ...
2
votes
1answer
63 views

Analysis of $f(x,y,z)=\frac{\sin(xyz)} {x^2+y^2+z^2}$

$f(x,y,z)=\frac {\sin(xyz)} {x^2+y^2+z^2}$ or $0$ if $(x,y,z)=(0,0,0)$ The problem says the following: a) Where is this function continuous, and b) Where is it differentiable? At a quick glance, the ...
1
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2answers
49 views

minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
1
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1answer
26 views

How to solve the limit $\lim_{(x, y, z) \to (0, \sqrt{\pi},1)}{e^{xz}\cos{y^2}-x}$

I started to solve this limit: $$ \lim_{(x, y, z) \to (0, \sqrt{\pi},1)}{e^{xz}\cos{y^2}-x} $$ But I'm confused on how keep going, I thought in convert the limit with spherical coordinates: $x = ...
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0answers
9 views

Integrating a function over a vector, involving linear projections of this vector

I have a function of the form $e^{w^{T}x}/(1+e^{v^{T}x})$ where $x,v,w$ are $n$ dimensional vectors. I'll be integrating this over $x$ , ie. $\int...\int\int e^{w^{T}x}/(1+e^{v^{T}x})dx_1dx_2...dx_n$ ...
1
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3answers
56 views

How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?

Problem. Let $f:[0,b]\times[0,b]\to\mathbb{R}$ be continuous. Prove that $$\int_0^b\left(\int_0^xf(x,y)\;dy\right)\;dx=\int_0^b\left(\int_y^bf(x,y)\;dx\right)\;dy.\tag{1}$$ My first thought was ...
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0answers
20 views

Solving differential equation when $x=0$ for $u$ when $u \,\,{du} = \frac{-k}{mx^2}dx$ where $u =u(t)=dx/dt$, $u(0) = 0$ and $x(0) = x_0$ and $x_0 >0$

The differential equation is following: $$u \,\,du = \frac{-k}{mx^2}dx$$ where $u =u(t)=dx/dt$, $u(0) = 0$ and $x(0) = x_0$ and $x_0 >0$. $k,m,x_0$ are positive constants. How do you solve this ...
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2answers
73 views

How do you maximize a polynomial over an integer domain?

I am working on maximizing the polynomial $f:\mathbb{R}^N \rightarrow \mathbb{R}$ $$f(v):=\prod_{i=1}^{N}( v_i+\alpha_i^2)$$ over integer $n$-partitions of $P$, $n\leq N$: $$\left\{v \left|\, v_i \in ...
1
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1answer
15 views

Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function. Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, ...
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2answers
24 views

Resolving MultivarLimit $3x^2y/(x^2+y^2)$ when $(x,y)\to(0,0)$ by approach

I'm resolving the limit of $\frac{3x^2y}{x^2+y^2}$ when $(x,y)\to(0,0)$ We didn't study any specal theorem, we did only approach. I tried first the suggested changes $y=mx$ and $y=x^2-x^3$. In ...
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1answer
19 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
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1answer
66 views

If the partial derivatives are $0$ is a function constant?

I am trying to prove that if we have a differentiable function: $f:\mathbb{R}^2\rightarrow \mathbb{R}$, and the partial derivatives of f is 0, then f is constant on a connected set. I am using the ...
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1answer
31 views

Why don't we go beyond the Hessian in multivariate optimization?

In univariate optimization, we perform the first derivative test to identify stationary points and the second derivative test to classify the stationary points as minima, maxima and inconclusive. When ...
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1answer
13 views

Finding explicit terms for $U(x,y)$ and $V(x,y)$

We are given $u = U(x,y)$ and $v = V(x,y)$ and $x = e^u \cos v$ and $y = e^u \sin v$. We are to find explicit terms form $U(x,y)$ and $V(x,y)$. I have tried some bruteforce-techniques, but I am ...
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1answer
31 views

How to determine the equations of the planes? [on hold]

Determine the equations of the tangent planes to the $x ^ 2 + y ^ 2 - 2xy + 4xz = 4 $ surface such that their normals are parallel to the line containing the points $P(2,1,4)$ and $Q(3,5,7)$.
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1answer
11 views

What is the maximum error?

Three positive numbers, each less than 100 are rounded to the first decimal place and then multiplied together, using differential, how I estimate the maximum error?
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2answers
33 views

Why is a vector function not smooth if $r'=0$

It is stated that a vector function is smooth if its derivative is continuous and nowhere zero, but I can’t find a proof or a geometric interpretation. Is this a definition or a theorem?
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1answer
86 views

Shape operator of the sphere.

I want to compute the Weingarten operator (shape) for the sphere $\{(x,y,z) \in \mathbb{R}^3 \ : \ x^2 + y^2 + z^2 = 1\}$. I am given the adapted frame: $$\left\{\begin{array}{l} E_1 = \cos \varphi ...
1
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2answers
31 views

Form of Function of Two Variables

Let $V(Q,T)$ be a function of two variables. The exact functional dependence is not known, but it is known that: $$V(Q,T)=f(Q)T,$$ and $$V(Q,T)=g(T)Q.$$ How do I prove rigorously that $$V(Q,T)=cTQ,$$ ...
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2answers
21 views

Lagrange Multipliers to determine min and max

I've got this question in a book of questions I'm doing. Can someone show me step by step how to solve this? Using Lagrange Multipliers for two constraints, determine the maximum and minimum of ...
0
votes
1answer
25 views

Multiple integral 3 dimension

Find the volume of the body $$ v:{(x,y,z) :\quad x^2+y^2\le z \le \sqrt{2-x^2-y^2}}.$$ I really don't know what to beside that i have to do triple integral of one. My main problem is to ...
0
votes
1answer
15 views

How to find a function with two variables from two functions with one variable

I am trying to determine a function for an algorithm I wrote. The time $t$ it takes to run depends on two variables $w$ and $l$ (with $l > 0$ and $w > 0$) I measured $t$ with a fixed $w$ ...
2
votes
2answers
31 views

A Lagrange Multiplier Problem : How to deal with this case when $b< 8$

I was trying to solve the following problem of several variables calculus given in my class.I am stuck in a particular case of the problem.Please help me to solve the problem.Thnx in advance. Find ...
0
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0answers
22 views

Partial derivatives using chain rule

If $u = \frac{1}{y}[\phi(ax + y) + \phi(ax - y)]$, and $\phi$ is twice differentiable, show that, $$\frac{\partial^2u}{\partial x^2} = \frac{a^2}{y^2}\frac{\partial}{\partial y}\left(y^2 ...
0
votes
2answers
46 views

Find a vector parallel to the intersection of the planes $2x-3y+5z=2$ and $4x+y-3z=7$

The solution is $(4,26,14)$. I know how to find the intersection of the planes, but not a parallel vector.
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0answers
24 views

Find the partial derivative with respect to y of the function $f(x,y)=ye^{xy}$

My solution was $e^{xy} + xy e^{xy}$, but when I checked the solution manual it said the answer is $xy e^{xy} \log e + e^{xy}$. So I solved each function for $y$ by setting them each equal to $0$. ...