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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24 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
0
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0answers
7 views

Certain local inequality for volume and surface measures

Suppose $S$ is closed simple piecewise smooth curve for in the plane (It is viewed as boundary of a domain). Does the following hold ...
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0answers
18 views

Calculating the magnetic vector potential

I've calculated A as proportional to r^2 for ra but I really don't think is correct. If someone could take me through the calculation I would really appreciate it.
0
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1answer
27 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
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1answer
33 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
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0answers
14 views

Using polar coordinates in this integral

I'm trying to solve something along the lines of: $$\iint \frac{\partial F_1(x,y)}{\partial x}+\frac{\partial F_2(x,y)}{\partial y}dydx$$ which I want to change to polar coordinates, but I don't ...
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2answers
38 views

Volume by double or triple integral?

I was trying to integrate the volume of a body blocked by $z=0,\; z=2x,\; x+y = 3$ and $y=0$ using the double integral... however it didn't work yet. I'm convinced its a double integral and not a ...
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1answer
18 views

Trying to integrate the volume of a body

I was trying to integrate the volume of a body blocked by $z=0$, $z=2x$, $x+y=3$ and $y=0$ using the double integral... but I don't really know how to approach this.
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0answers
31 views

Integration of a symmetric function

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...
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0answers
13 views

Proving that average value of $u$ around a circle is the value of $u$ at the centre.

I would like to prove that: If $u(x,y)$ is harmonic in a domain containing a disk of radius $r$ with boundary $C_r$ $\implies$ the average value of $u$ around the circle is the value of $u$ at the ...
5
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1answer
347 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 ...
2
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1answer
35 views

Trying to integrate $\iint_D x^4\tan(x)+3y^2 \,dA$.

I'm trying to integrate $\iint_D x^4\tan(x)+3y^2\, dA$ in domain $D=\{(x, y) \in \Bbb R^2 \mid x^2+y^2\le4, y\ge0\}$. Domain is simple enough; half circle of radius 2 over $x$ axis. Converting to ...
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3answers
52 views

Area of a region under the mapping $f$

Consider the function $f:\mathbb R^{2} \to \mathbb R^{2}$ given by $f(x,y)=\left(e^{x+y},e^{x-y}\right)$. Area of the image of the region $\{(x,y)\in \mathbb R^{2} | 0<x,y<1\}$ under the mapping ...
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0answers
20 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
2
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0answers
52 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
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3answers
48 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
21
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5answers
13k 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 easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to ...
0
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0answers
11 views

formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
1
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2answers
21 views

Plot the level curve of sine function in multiple variables

I'm very confused about how I could go about this, as it seems that the question cannot be done using only the information given. The question is: plot the level curve for $f(x,y) = \sin(k^2x^2 + ...
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2answers
37 views

Change of variables in multivariable differential equations

This is a very easy question about how to justify the change of variables. Let $f$ be a $C^1$ function of two variables $x,y$. Introduce the variables $s,t$ as: $$\begin{cases} s=x+y \\ t=x-y ...
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0answers
23 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
2
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3answers
64 views

Finding Extrema of $f(x,y)=x^4+y^4-4xy$

Let $f(x,y)=x^4+y^4-4xy$ How do I find all the relative extrema and saddle points of $f$ which lie within the open square ${(x,y) | -2<x<2,-2<y<2}$. And also if $f$ was in the closed ...
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0answers
24 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
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1answer
48 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
1
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2answers
50 views

Are partial derivatives a special case of the total derivative or just something else entirely?

I can do basic multivariable calculations using partial and total derivatives. I also know for partial derivatives the existence of all partial derivatives at a point doesn't imply continuity. Are ...
1
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2answers
40 views

Find a unit vector that is parallel to $\nabla f(\cos\theta,\sin\theta)$

Suppose $f(x,y)$ is differentiable for all $(x,y),f(x,y)=17$ on the unit circle $x^2+y^2=1$, and $\nabla f$ is never zero on the unit circle. For any real number $\theta$, I have to find a unit vector ...
1
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0answers
9 views

Finite cover of balls for a bounded subset

Suppose X is a bounded subset of a k-dimensional manifold $M$ $\subset$ $R^n$ with $(k-1)$-dimensional volume $0$. I need to show that this implies for all $d>0$, all points of X can be contained ...
0
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0answers
34 views

surface integral of the normal component of $curl(\mathbf{F})$

I have a square with side lengths $a$ in the $xy$-plane, and we build he square up vertically in the $z$-direction to construct a cube. What is the surface integral of the normal component of the ...
1
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2answers
30 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
1
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1answer
24 views

Finding domain, range, and level curves of $f(x,y)=\arcsin(6y-5x)$

I will like it if someone helps me out and also checks my work for me for this equation: $f(x,y)=\arcsin(6y-5x)$ Domain $-1\leq6y-5x\leq1$ $y\geq\frac{5x}{6}-\frac{1}{6}$ and ...
2
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1answer
40 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...
2
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1answer
35 views

$f(x,y)=4x^3y^2$ Directional Derivative…

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
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0answers
17 views

$f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
1
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0answers
18 views

Volume of the solid between these two parabaloids - can someone verify my answer?

I wish to find the volume of the solid bounded by the surfaces: $z = x^2 + y^2$ and $z = 4 - a^2x^2 - a^2y^2$. I set the two surfaces equal and it gave me a circle, so, I used cylindrical ...
0
votes
1answer
28 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
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2answers
55 views

Why does $\frac{\partial F}{\partial x_i}(\pmb{a}) = \lim_{h\rightarrow 0} \frac{F(\pmb{a}+h\pmb{e}_i)-F(\pmb{a})}{h} $ gives a vector?

I am reading about partial derivatives in some notes on Multivariable Calculus. On page 2, it says: Definition: Partial Derivatives Let $U \subset \mathbb R^m$ be open, let $F: U \to \mathbb ...
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0answers
23 views

Computing $f_{xy}$ and $f_{yx}$ [duplicate]

Let's consider the following function: $$f(x,y)=\begin{cases} xy\left(\dfrac{x^2-y^2}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I need to compute ...
0
votes
2answers
27 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
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1answer
55 views

Proving that $f$ is differentiable at $0$

Let's consider the following function: $$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I know that ...
0
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0answers
35 views

Find Directional Derivative, Unit Vector, and Rate of Change

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
3
votes
1answer
25 views

Lagrange multipliers method - absolute maximum and minimum

Using the Lagrange multipliers method I have to find the absolute maximum and minimum value of $f(x, y)=x^2+y^2-x-y+1$ in the unit disc. So, I have to find the extremas of $f(x, y)=x^2+y^2-x-y+1$ ...
2
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2answers
57 views

How to evaluate $\sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty \frac{1}{j^2-k^2}$

I was reading an introductory text on multiple integrals and I have encountered a problem asking me to explain why $$ \sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty ...
2
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1answer
55 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
2
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1answer
18 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
9
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2answers
3k views

Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
0
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1answer
623 views

Equation of plane through intersection of planes and parallel to line

Find the equation of the plane through the intersection of the planes of $x-2y+z=1$ and $2x+y+z=8$ and parallel to the line: $\frac{x-3}{1} = \frac{y-1}{2} = \frac{z-2}{1} $ I'm facing difficulties ...
0
votes
1answer
20 views

Parameterise the path C of a square

I have a question, I am required to parameterise the square with side lengths $a$, going in a counterclockwise direction. I have determined then that the points are $$(0,a), (a,0), (0,0), (a,a)$$ ...
2
votes
1answer
48 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
6
votes
0answers
62 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
1
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0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...