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
votes
2answers
57 views

Need help with Double Definite Integration

I need help in solving this double integral: $$\int\limits^2_{-2}\;\;\int\limits^\sqrt{4-x^2}_{-\sqrt{4-x^2}}{(x^2+y^2)^{5/2}}dy\,dx$$ Maybe introducing polar coordinates might help?
2
votes
1answer
77 views

Volume of Solid Defined by Inequalities

How can I find the volume of a solid defined only by inequalities? For example, in this case I have: $$0\le z \le y \le x \le 1$$ Can someone please explain to me step-by-step on how I can do this. ...
1
vote
1answer
54 views

Finding Surface area of Paraboloid [duplicate]

I am having trouble finding the surface area of the part of the paraboloid that lies in the first octant of $z=5-x^2-y^2$. So, I realized that the first octant refers to when, $x,y,z \ge 0$. How do I ...
0
votes
1answer
17 views

Use lagrange multultipliers to find the indicated extrema

maximize $f(x,y,z)=x+y+z$ subject to $x^2+y^2+z^2=1$ I do not understand this at all or where to go from here would appreciate some insight
0
votes
1answer
33 views

Find the partial deriavtive with respect to x and y

$$f(x,y)=\ln \frac{x-y}{(x+y)^2}$$ Use log properties I started with this $$\ln(x-y)-2\ln(x+y)$$ I got this for $x$: $$\frac{1}{x-y}-\frac{2}{x-y}$$ I got this for $y$: ...
-1
votes
3answers
50 views

How to generalize $f = \frac{u}{v}$, $u,v$ scalars, for vectors? [closed]

Suppose $f = \frac{u}{v}$ for some scalars $u,v$. How does one go about generalizing this for vectors $\mathbf{u,v}$? I think there is no concept such as division by a vector ..
1
vote
0answers
60 views

When can I calculate a derivative in a point?

Okay the title makes no sense. I have a two variable function, $f(x,t)$. When is it that $$ \left(\frac \partial{\partial x} f(x,t) \right)\bigg| _{t=0} = \frac{d}{dx} f(x,0)$$? My guess is that it ...
1
vote
1answer
35 views

Difference between a Fréchet derivative and a total derivative

I've heard many times that they are somehow similar and in some cases mean the same thing. Consider this function: $$f(x,y)=x^2y$$ I have to calculate the Fréchet derivative $f'(x_0,y_0)$ and some ...
1
vote
1answer
37 views

Diffeomorphism which has a zero

Let $f:B(x_0,r) \subset \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism between $B(x_0,r)$ and its image. If $|f'(x)^{-1}| \leq M$ for all $x \in B(x_0,r)$ and $|f(x_0)|<r/M$, show that ...
2
votes
2answers
164 views

Does the concept of a derivative a rate of change work for n dimensions?

I am trying to understand what exactly a derivative is. I understand the total derivative is a linear map. But I don't understand what happens to the idea of a rate. In high school calc, one is ...
0
votes
1answer
20 views

computing flux and circulation using Green's theorem

I must compute the outward flux and counter clockwise circulation of $F$ through and around $C$ using Green's theorem. $F=<xy,\:x+y>$, ...
0
votes
1answer
28 views

Equilibrium Points for 8th Degree Polynomial

I have an 8th degree polynomial that I need the zeros for. Is there even a way to explicitly solve one? Its for a diff equations review. I need to sketch the phase line, which is a breeze once I get ...
3
votes
1answer
174 views

How can I solve $\lim_{(x,y) \rightarrow (0,0)} \frac{xy\sin(x+y)}{x^2+y^2+|xy|}$?

How can I solve: $$\lim_{(x,y) \rightarrow (0,0)} \frac{xy\sin(x+y)}{x^2+y^2+|xy|}$$ I used Wolfram Alpha, but it said that it doesn’t exist. If I do it by myself I get: $$0 < ...
1
vote
0answers
33 views

A question about a system of PDE

It is well known that under suitable conditions, the symmetry of mixed second partial derivatives reads: $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}.$$ ...
0
votes
3answers
56 views

Can I argue that $g'$ is non zero in this case?

Consider two smooth maps $g,f$ given by $$ {\partial \over \partial x} g(x)= g'(x) = \int_0^1 {\partial \over \partial x} f'(u + t(x-u)) dt = \int_0^1 f''(u + t(x-u)) \cdot t dt $$ where $f' = ...
2
votes
1answer
33 views

Iterated Integral and Sign Change in Answer

Given the iterated integral $\int_0^1\int_x^{2-x}(x^2-y) \, dy \, dx$, the value for the type I integral is, \begin{align*} & \int_0^1\int_x^{2-x}(x^2-y)\,dy\,dx \\ = {} & \int_0^1 ...
0
votes
0answers
42 views

Change of variable and partial derivative

Let us suppose we consider the following change of variables $(t,r)\rightarrow (T,R)$ with $$ f(t,r) \frac{\partial}{\partial t}= \frac{\partial}{\partial T} \quad(1)\\ g(t,r) \frac{\partial}{\partial ...
0
votes
1answer
26 views

Compute Surface Integral

Integrate $x^2+y^2$ over the upper hemisphere of radius $a>0$ centered at $(0,0,0)$. $\textbf{Edit}$ Consider the parametrization of the upper hemisphere given by $$X(\phi, \theta) = (a ...
0
votes
2answers
34 views

Show the function is continuous in $\Bbb R^2$

Show that the function f is continuous in $\Bbb R^2$ : $$f(x,y)= \left\{ \begin{array}{ 1cc} \sin(xy)/xy & xy\neq0\\ \\ 1, & xy =0 \\ \\ ...
2
votes
1answer
36 views

How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
-1
votes
2answers
41 views

Continuous derivative vs Continuous partial derivatives

Firstly, suppose $f:\mathbb R^n\to\mathbb R^m$ has all continuous partial derivative. I believe I have proved that this imply continuous derivative. Please tell me if this is actually true. For its ...
2
votes
0answers
33 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
1
vote
1answer
31 views

How to prove this identity in vector calculus (suffix notation)?

Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or ...
0
votes
1answer
33 views

$\frac{\partial u}{\partial x} \cos \theta + \frac{\partial u}{\partial y} \sin \theta = \frac{\partial u}{\partial r} (z_0 + r e^{i\theta})$?

Let $u$ be a function of two variable and all its partial derivative exists and fix $z_0 \in \mathbb C $ and $r>0.$ My vague question: How to show: $\frac{\partial u}{\partial x} \cos \theta + ...
0
votes
1answer
13 views

Curvature at a point in a vector valued function

I am trying to determine the curvature when $t=2$ of the function $r(t)=<t^3,3t^2,8t>$ So I found $v(t)=<3t^2,6t,8>$ and $a(t)=<6t,6,0>$. So now that I have these two functions, I ...
1
vote
2answers
50 views

Minimizing a summation?

I have absolutely no idea how to approach this problem. I've been looking through notes, and I think I missed this when my professor discussed this in class. $$ \text{Consider the data}\\ i\: x_i\: ...
1
vote
4answers
115 views

Derivative of $\pmb{A}^T\pmb{x}$ with respect to $\pmb{A}$

Let $\pmb{A}$ be a real matrix and $\pmb{x}$ a vector such that $\pmb{A}^T\pmb{x}$ exists. Then how do I calculate the following result? $$\frac{\partial}{\partial \pmb{A} } \pmb{A}^T\pmb{x} = ?$$ Any ...
-1
votes
1answer
64 views

Continuity of a function on $\Bbb R^2$ [closed]

Function $f(x,y)$ is defined in a neighborhood of $(0,0)$. Then if for any t function $g(x) = f(x,tx)$ is continuous at $0$, then $f$ is continuous at $(0,0)$. if $f$ is continuous at ...
2
votes
3answers
39 views

Find maximum and minimum values of an equation on an elipse

I need some help with this. I've been struggling through this last chapter of my Calc III class, and I'm not sure how to do this (although, it doesn't seem like it should be difficult to do) $$ ...
-1
votes
1answer
27 views

Limit of a multidimensional function

Limits problem again... Is there an easy and fast way to calculate $$\lim_{(x,\,y)\to 0} \frac{\sin(x)}{y}$$ and $$\lim_{(x,\,y)\to 0} \frac{\sin(x)}{y} \cdot \frac{1}{||(x,y)||_2}?$$ Thanks!
2
votes
0answers
16 views

Existence of full-rank function with given components

Assume that $U \subseteq \mathbb{R}$ is open and $f_1,...,f_k : U \rightarrow \mathbb{R}^n$ are differentiable functions such that $(f_1,...,f_k)$ has rank $k$ everywhere. Can we find more functions ...
0
votes
1answer
33 views

Differentials of Multivariable Functions

A soft drink can is h centimeters tall and has a radius of r cm. The cost of material in the can is 0.0015 cents per cm$^2$ and the soda itself costs 0.002 cents per cm$^3$. The cans are currently 10 ...
3
votes
1answer
43 views

Inverse Function Theorem, Spivak's Proof

I'm having a lot of trouble following the proof of the following theorem. This is from Spivak's Calculus on Manifolds. 2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n \to ...
2
votes
2answers
45 views

$\nabla \times \left(\frac{\mathbf{A \times r}}{r^3}\right)$, where $\mathbf{A}$ is independent of $\displaystyle\nabla \times$

The curl is just over $\mathbf{r}$ and $r$. I've been trying to pull the vector $\displaystyle \mathbf{A}$ out of the way, in order to get a expression much easier to deal with, but I have no idea how ...
1
vote
2answers
39 views

The derivative of a recurrence relation of functions

I am unsure of how to take the derivative of a recurrence relation of functions. For example consider the following recurrence relation: \begin{equation} \left\{ \begin{array}{cl} f_n(x) &= ...
0
votes
1answer
41 views

Volume of a Solid Bounded by Surfaces and Planes [closed]

How do I find the volume of a solid that is bounded by surfaces: $$z=e^{x-y}, z=-e^{x-y}$$ And the planes: $$x=0,y=1,x=y$$ My thoughts: The first thought that came to mind was double integrals.
3
votes
3answers
92 views

Points on Surface, Distance Optimized

How do I find the points on the surface: $$x^3+y^3+z^3=1$$ such that the distance to the origin is minimized? My Thoughts: Perhaps we can minimize the distance squared? Not sure.
2
votes
1answer
58 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
2
votes
0answers
41 views

Extensions of $C^k$ functions to the boundary [closed]

Assume $\Omega \subset \mathbb R{^n} $ is an open connected smooth domain. I have some propositions that I guess they are correct , but I want to be confident. If $f\in C_0^k(\Omega) $ then $f\in ...
2
votes
1answer
30 views

How to express curvature of a level set in terms of derivatives of a function?

Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) ...
0
votes
2answers
17 views

Fidning multivariable limit

I'm having trouble in finding multi-variable limit and hope that an example like this could get me started on my work $$\lim \limits_{(x, y) \to (2,0)} \frac{1-cosy}{xy^2}$$
2
votes
2answers
33 views

$\displaystyle\iiint_E (x²+y²) \;\mathrm{d}V$ where $E$ is the region between the spheres $x^2+y^2+z^2 = 4$ and $x^2 + y^2 + z^2 = 9$

To be honest I'm not even too sure of what I'm integrating. I'm picturing two spheres touching each other, with a cylinder of two different radii going from the center of one to the other and I'm ...
0
votes
2answers
27 views

Derivative of dot product?

What's the derivative ${\partial \over \partial x} \langle x, f(x)\rangle$? According to the product rule it should be $1\cdot f(x) + x \cdot f'(x) $ but in my previous post I was told that this ...
1
vote
1answer
48 views

Any real valued continuous function from a closed bounded set in $\mathbb R^2$ is bounded and attains its bounds

Without using the idea of compact or sequential compactness , can we prove that if $A$ is a closed bounded set in $\mathbb R^2$ , then any continuous function $f:A \to \mathbb R$ is bounded and ...
0
votes
2answers
32 views

Critical points of $(x^2+y^2)e^{y^2-x^2}$

$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$ $\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$ So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$. $x(1-x^2-y^2) = y(1+x^2+y^2)$ $x-x^3-xy^2 ...
0
votes
1answer
33 views

Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0.

The problem from Munkres' *Analysis on Manifold is that Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0. My thought on the first ...
2
votes
3answers
64 views

Help finding specific book

I'm studying Engineering and I'm in my second year, studying Multivariable Calculus, but my University is kind of hard teaching me fresh calculus with topology and analysis, and is kind of hard, so I ...
1
vote
0answers
25 views

Vector Differential in specific form

for the operator: $\Delta A(t) = A(t+1)-A(t)$, let : $$ \Delta C = \sum^{T-1}_{t=0} [~~H(\pi(t+1)~|~\mu(t+1))~~ - ~~H(\pi(t)~|~\mu(t+1))~~] $$ Where $H (\pi(t+1)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log ...
0
votes
2answers
34 views

Taking derivative of function $g: \mathbb{R} \to \mathbb{R}$ defined in terms of $f: \mathbb{R}^{n+1} \to \mathbb{R}$.

Suppose we are given $g(r): \mathbb{R} \to \mathbb{R}$ where $g(r) = f(ry, r^2s)$ for $f: \mathbb{R}^{n+1} \to \mathbb{R}$ where $y \in \mathbb{R}^n, s \in \mathbb{R}$. How do we determine ...
1
vote
3answers
51 views

Critical points of $f(x,y) = \sin x \sin y, -\pi<x<\pi, -\pi<y<\pi$

$\frac{df}{dx} = \cos x\sin y = 0$ $\frac{df}{dy} = \cos y\sin x = 0$ $\cos x\sin y = \cos y\sin x$ $\frac{\cos x}{\sin x} = \frac{\cos y}{\sin y}$ $\cot x = \cot y$ $P_1 = (\pi/2,\pi/2)$ $P_2 = ...