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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43 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
2
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0answers
22 views

Green's Theorem proof using Stoke's theorem

I'm slightly confused with this proof, from Stoke's Theorem we have: $$\int_C \underline{F} \cdot \ d \underline{r} = \int \int_S (\nabla \times \underline{F}) \cdot \underline{n} \ dS$$ so going ...
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2answers
60 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
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1answer
25 views

particular solutions is zero

We have some issues with particular solution. Cannot solve A and B on the last line because it becomes zero all together. So it becomes $2sin(2x) = 0$ What are we doing wrong? Thanks for your time. ...
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0answers
40 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
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0answers
33 views

Chain rule for several variables

I am studying this example: I follow the first two statements, but I cannot make the connection between the dot product and the derivative. Can somebody please explain how the third equation ...
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0answers
14 views

maximize the acos function of more than one variable

I want to find the maximum angle which is defined as follows: $\theta = \cos^{-1}(\frac{1 + x_1 + x_2}{\sqrt{1 + x_1^2 + x_2^2}})$ now I should find $x_1$ and $x_2$ values so that $\theta$ has its ...
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0answers
30 views

Partial Derivative With Respect to $t$

What is $\frac{\partial v}{\partial t}$ if $v$ can be defined as $v(x,t,\zeta)=w(x(3t)^{-1/3},\zeta (3t)^{1/3})$?
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21 views

Total derivative

What is the significance and meaning of the total derivative? Why is it introduced in the definition of differentiability of scalar and vector fields?
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0answers
13 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
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0answers
8 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
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6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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0answers
12 views

How to set up the surface integral?

Find the surface area of the piece of the cylinder x^2 + y^2 = 4 cut off by the planes z = 0 and y = z with y greater than or equal to 0 using surface integrals. Can someone help me set up this ...
3
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0answers
28 views

Multivarable limit proof

I came across with this statement and I can't neither prove it right nor find a counterexample. The statement is: Consider two functions $F(x,y)$ and $G(x,y)$ continuous and differentiable around a ...
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1answer
41 views

In this problem, what am I taking the integral of?

I'm a little confused by this problem because I have no idea what I should be taking the integral of. I was following a book example when I realised that the book explicitly tells you what to take the ...
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2answers
27 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
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2answers
18 views

Calculating partial derivative for a function defined by an integral

I have no idea how to solve the following problem. Please suggest some suitable solutions. Define $$f(x,y)=\int_0^{\sqrt{xy}} e^{-t^2} \,dt,$$ for $x>0, y>0$. Compute $\dfrac{\partial ...
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5answers
1k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
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0answers
25 views

A question on gradients: $f$ must assume equal value at two points

If $\nabla f(x,y,z)$ is always parallel to $x i+y j+z k$, show that $f$ must assume equal values at the points $(0,0,a)$ and $(0,0,-a)$.
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1answer
29 views

k+1 Differential form

Consider the k-form given by, $ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
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1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
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0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
2
votes
1answer
58 views

Verify Stokes's Formula for…

Verify Stokes's Formula for $\textbf{F}(x,y,z)=(3y,-xz,yz^2)$, where $S$ is the surface of the paraboloid $2z=x^2+y^2$ bounded by the plane $z=2$. So I need to compute the integral using the formula ...
2
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1answer
39 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
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0answers
8 views

Basic cartesian to cylindrical coordinate conversion?

I have solved the conversion for all of them except for the one highlighted. I have tried numerous solutions and still cannot get the correct answer. Solving for this in the coordinate, I did ...
3
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2answers
39 views

Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...
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0answers
19 views

Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
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2answers
18 views

Application of chain rule

The equations $u=f(x,y),x=X(t),y=Y(t)$ define $u$ as a function of $t$, say $u=F(t)$. Compute $F'(t)$ in terms of $t$ if, $$f(x,y)=\log [(1+e^{x^2})/(1+e^{y^2})] , X(t)=e , Y(t)^t=e^{-t}.$$ From the ...
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0answers
14 views

Multivariable differentiation under the integral sign

Suppose that the functions $f:[a,b]$x$[a,b] \to \mathbb{R}$ and $\frac{\partial{f}}{\partial{t}}:[a,b]$x$[a,b] \to \mathbb{R}$ are continuous. Prove that the function $F:[a,b]$x$[a,b] \to \mathbb{R}$ ...
1
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1answer
20 views

Understanding optimization on non-compact region

Say we have $f(x,y) = x^2 e^{-x^2 - y^2}$ and we want to optimize it over $\mathbb{R}^2$. The minimum value is $0$ since $f(x,y) \geqslant 0$; the question is whether a maximum value exists or not. ...
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2answers
21 views

Polar coordinates and Jacobian of $\frac12 r $

To solve a double integral problem, I just did the sub $$x = \frac12 r \cos( \theta ) , \quad y = r \sin( \theta )$$ and the Jacobian is $\frac12 r $ but I realise – I'm not sure how to write that ...
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5answers
271 views

Why aren't these partial derivatives interchangeable?

I've ran across something that confuses me regarding multivariable functions and partial derivatives. I'll use an example to illustrate: We let $$x = f(y,t) = yt^2,$$ and define the operators ...
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2answers
67 views

Prove that $f'_{xy}=f'_{yx}$ [duplicate]

Here is a basic, and probably a bad, question. A fundamental rule of derivatives. Why is $f'_{xy}=f'_{yx}$ true?
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0answers
16 views

Nonexistence of a scalar field

Prove that there is no scalar field $f$ such that $f'(a;y)>0$ for a fixed vector $a$ and every non-zero vector $y$.
1
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1answer
25 views

Constant function on a convex set

If $f'(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $R^n$, prove that $f$ is constant on $S$. A set $S$ is called convex if for every $a$ and $b$ in $S$, ${ta+(1-t)b \epsilon S}.$ ...
2
votes
1answer
38 views

What is the difference between parametrization and change of variables?

I am embarrassed to ask this, but really need to, in order to clarify my confusion. I am taking multi-variable calculus and I am confused as to the difference between when I should be parametrizing ...
3
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1answer
25 views

Chain rule quesition: proving that the Weingarten map is self-adjoint

I'm reading through the proof in this paper (http://www.math.leidenuniv.nl/scripties/JaibiBach.pdf) but I'm stuck at the line: "Using the chain rule we get: $L_p(\phi_v) = -Dn(\phi_v) = - \frac ...
2
votes
1answer
26 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
2
votes
1answer
37 views

Applying the Implicit Function Theorem - how to evaluate the partial derivatives that arise?

The problem: We have a function $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ such that $f(x,y,z) = (x-xy, x+2y+z^2)$. For a point $(a,b,c)$ such that $f(a,b,c) = 0$, find a condition on $(a,b,c)$ such ...
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0answers
42 views

How to solve the following problem?

I want to obtain the result of this derivative. if z isn't fixed at 5, I know we can solve this through euler lagarange equation.
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0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
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0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
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1answer
30 views

Volume using spherical coordinates-finding theta!

Let $V$ be the region in $\mathbb{R}^3$ defined by the inequalities $z^2\ge x^2+y^2$, $x^2+y^2+z^2\leq 1$, and $z\ge 0$. Sketch the region $V$ and find its volume? I have sketched $V$ but to find the ...
3
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0answers
33 views

How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
0
votes
1answer
26 views

Integral over asteroid

Calculate $$\int_L\frac{x^2dy-y^2dx}{x^{5/3}+y^{5/3}}$$ where $L=\{x^{2/3}+y^{2/3}=R^{2/3}\}$ (and L is part of an asteroid as far as I know).
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2answers
20 views

Maximize the directional derivative

Find the points $(x,y)$ and the directions for which the directional derivative of $f(x,y)=3x^2+y^2$ has its largest value, if $(x,y)$ is restricted to be on the circle $x^2+y^2=1$. For the point ...
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2answers
50 views

Triple integral - getting the region right

I have this problem: $$\iiint_D xyz \, dx \, dy \, dz,$$ with $D=\{(x,y,z) : 0 \le x \le y \le z \le 1 \}$. I tried solving it the same way, only that I did $y$ from $x$ to $z$, and not $0$ to ...
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1answer
40 views

Directional Derivative and differentiability

My question is similar, but not equal to this...Question on linearity of directional derivative Let $f'_{h}(a)$ be the directional derivative. And for the function $f:\mathbb{R}^n\rightarrow ...
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2answers
30 views

Parametrisation of a surface.

I'm currently going through my course notes, currently teaching the theory leading up to surface integrals. In particular, I am reading the section on the parametrisation of surfaces. However, there ...
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2answers
24 views

Finding a partial derivative of a double summation. [closed]

How to find $$\frac{\partial T }{ \partial {\dot q_i}},$$ given that $$T= \sum_i\sum_j{\alpha_{ij}\dot q_i\dot q_j}?$$