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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25 views

A calculus problem regarding mass

I am reviewing calculus and working on the following problem. In $\mathbb{R}^{3}$, the density function is given by $\mu(x, y, z) = |z|$ and if the region $R$ is given by $R : 2 \leq x^{2} + y^{2} + ...
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1answer
66 views

An example in Spivak's Calculus on Manifolds (chain rule).

Spivak gives an example which has step that is giving me some problems to get it, even if it's supposed to be trivial. Spivak says: Let $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(x,y)=\sin(xy^2)$. ...
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1answer
40 views
+50

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
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19 views

Integrals depending upon a parameter

There was an exercise, in my professor's book, asking to prove the continuity of an integral depending upon a parameter. Namely, the hypothesis were: Let $D$ be a measurable subset of $\mathbb{R}^n$, ...
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21 views

Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case?
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1answer
33 views

How to use Differentials to find the area of a ring? [duplicate]

A ring of a planet has an inner radius of approximately 51,400 km (measured from the center of the planet) and a radial width of 17 km. Use differentials to estimate the area of the ring. (Round your ...
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1answer
36 views

Show $\iint xye^{-xy}\,dx\,dy$ is convergent or divergent

Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant. I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other ...
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1answer
28 views

Does the fact that we can parametrize surfaces rely on Fubini's theorem?

When learning about double integrals, we learn that the only way we can do them is if Fubini's theorem applies (or maybe I should say, the only way I learned to do them in that one class). But then ...
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33 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
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0answers
25 views

Calculate the workload of srs program

This is a real world problem I'm trying to figure out. This seems like calculus to me, but I didn't take calculus in college so I don't know where to look to solve my problem. I'd appreciate any ...
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0answers
25 views

Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Prove $\sigma$-additivity in the ff: Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} ...
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2answers
20 views

Line integrals; How to set $t$ boundary?

I'm having a hard time understanding how to set t boundaries in line integrals... The question is: find the line integral of $f(x,y,z)$ over the straight line segment from $(1,2,3)$ to $(0,-1,1)$. I ...
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2answers
87 views

What is the difference between a line integral with respect to x or y and a Riemann integral with respect to x or y?

I'm finding the concept of line integrals with differentials including dx or dy hard to swallow intuitively. Specifically, I'm having trouble differentiating them from a Riemann integral. What are the ...
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3answers
69 views

Evaluating the integral of an exact differential

What is wrong with evaluating the closed path integral as the following? $$ \oint_\gamma \frac{x\,dy-y\,dx}{x^2+y^2}= 2\pi\ne\oint_\gamma d\left(\arctan\left(\frac{y}{x}\right)\right)=0 $$ where the ...
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1answer
75 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
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1answer
22 views

$f(x,y,z)$ must be equal at $(0,0,a)$ and $(0,0,-a)$ when $\nabla f(x,y,z)$ is always parallel to $(x,y,z)$

This vector calculus exercise from Apostol's book has taken me some hours without any advance. Can you give some hints please? If $\nabla f(x,y,z)$ is always parallel to $(x,y,z),$ prove that $f$ ...
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3answers
43 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
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0answers
22 views

Non Linear general kinematic wave equation

I am rather confused by this section of my non-linear waves notes. In the parts I have underlined in green $c(u_0(\xi))$ is defined as a constant and then as a variable even though in both instances ...
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2answers
22 views

A rigorous statement and proof of the two-path test

Background In multivariable calculus/complex analysis, one often tries to show that a limit doesn't exist by finding two paths which yield different limits. The justification goes that if the limit ...
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1answer
24 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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1answer
23 views

Partial Differential from Implicit Expression

I have an expression which is explicit in $p$ given by ...
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1answer
31 views

What is a closed form expression for the ∂/∂w(∂t/∂w) if w(t) is complicated function?

Lets say we have a trigonometric function w(t) that can not be inverted as t(w). The derivative ∂t/∂w can be calculated as 1/(∂w(t)/∂t). What is a closed form expression for the second derivative ...
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1answer
57 views

Double integral over complicated region

Suppose we wanted to compute $\iint\frac {1}{1 + x^2 + y^2} dxdy$ over the region $\frac {(x-1)^2}4 + \frac {(y+2)^2}9 \leqslant 1$. It gets quite hairy if we use elliptical polar coordinates i.e. ...
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1answer
40 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
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1answer
25 views

Finding the image of a region transformed by a mapping

The only examples I've found are either very complicated, or state the transformation like y=g(u,v) x=f(u,v), whereas this question states u and v in terms of x and y. I'm not sure how to get ...
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2answers
45 views

Continuity of a multivariable function with “parts”

I'm trying to solve if $f$ is continuous: $$ f(x,y) = \begin{cases} x^3 + y^3 &\text{if }y>0 \\ x^2 &\text{if }y ≤ 0 \end{cases} $$ I have seen that $$\lim_{(x,y) \to (0,0)} ...
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1answer
18 views

Find an equation for the plane

Here is the whole question. Find an equation of the plane that passes through the points $P(1,0,-1)$ and $Q(2,1,0)$ and is parallel to the line of intersection of the planes $x+y+z=5$ and $x+y-z=1$. ...
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30 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
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1answer
22 views

Meaning of the Notation $\mathcal{F}[X,F(X)]$

I came across the following theorem. What is the meaning of the notation $\mathcal{F}[X,F(X)]$? Thanks, Jay.
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1answer
29 views

Particular solution of reducible to homogeneous equation

Verify that $y=x-5$ is a particular solution of the equation $$\frac{dy}{dx} = \frac{2y+6}{x+y+1}\ .$$ This is when $y'=1$ but this is not given as a condition in the question. How would you write the ...
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1answer
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35 views

Is there a formal proof for this theorem??

There is a theorem in the book Advanced Calculus by Wilfred Kaplan which states the following: The differential formula : $$ dz = \frac {\partial z}{\partial x} dx + \frac{\partial z} {\partial y}dy ...
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3answers
62 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
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1answer
28 views

Special form of the Divergence theorem

It states in my notes relating to the derivation of the euler equations that a 'special form' of the divergence theorem is: $\iint{\phi\hat{n}}\space{dS}=\iiint\nabla\phi\space{dV}$ with $\phi$ a ...
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2answers
72 views

Typed version of Newton's Principia Mathematica

I need a typed pdf version of Newton's Principia. Is it available for free online? And I also need the proof of universal law of gravity and the elliptical orbits of planets(If there's no typed ...
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2answers
32 views

Double integration:$ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $

I would be grateful for a little help if someone could help me solve a problem in my textbook. The question is, evaluate $ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $, where $a,b$ are positive ...
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1answer
20 views

Find the point of intersection of plan and parabaloid

Find the point of intersection of the plane $x+2y+z=10$ and the parabaloid $z=x^2+y^2$ that is closest to the origin. Do this by minimizing the distance squared from the origin: ...
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1answer
30 views

Find the absolute extrema by Lagrange multipliers

Find the absolute extrema for the function $g(x,y)=e^{x^2}-y^2$ on the unit disk $D$ given by: $D=\{(x,y)|x^2+y^2\le1\}$. Do this by first finding all critical points of and classifying them, then ...
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12 views

find the absolute extrema

Optimize the function f(x,y)=x^2y on the elliptical cylinder x^2+2y^2 =< 6 using Lagrange Multipliers well, from what I know that I have to find the gradient then to set it equal to zero but i'm ...
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3answers
56 views

What is the range of $f(x,y)=e^{-(x^2+y^2)}$

I know the domain is $\mathbb{R^2}$. Is the range $\mathbb{R}$?
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3answers
123 views

Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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1answer
20 views

Compute double integral on polar coordinates, find $r(\phi)$

I have the function $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ and the region $\{y\geq 2x^2-2, y\leq 3x\}$. The region is: To compute the integral in cartesian coordinates: ...
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1answer
24 views

The acceleration in terms of the eulerian velocity

I'm having trouble in deriving the the acceleration in terms of the eulerian velocity. How do I apply the chain rule for partial derivatives to achieve the result highlighted in green?
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22 views

Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
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1answer
32 views

Question involving double integral

How can I calculate the integral $$\iint_D \frac{\partial}{\partial y}\frac{y}{(x^2+y^2)^2} \, dx \, dy$$ where $D=\left\{ (x,y) \in \mathbb{R}^2 \mid 1\leq x^2+y^2 \leq 4 \right\}$ ? Thanks !
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2answers
51 views

Find the gradient of $\frac{x}{x-y}$

It seems simple on the face of it, but I cannot figure out how to actually do this. I know that you have to find the partial with respect to $x$ and also with respect to $y$, but that's where I get ...
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1answer
41 views

Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
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15 views

About integration limits over region

I need to integrate $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ over $\{y\geq 2x^2-2, y\leq 3x\}$ Im in doubt if it can be computed in just "one step" as: ...
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31 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
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22 views

Volume under surface

What is the volume under the surface $z = f(x,y) = x^4 + xy + y^3$ over the rectangle $R = [1,2] \times [0,2]$. I solved the double integral by integrating with respect to $x$ and then with respect ...