Questions tagged [multivariable-calculus]
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).
0
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0answers
13 views
Change of variable in double integrals.
Problem
Express the double integral $\iint_S f(x,y) \,dx \,dy$ as an iterated integral in polar coordinates,where
$S$={$ (x,y)|x^2 \leq y,-1 \leq x \leq 1$}.
Doubt
I have difficulty finding ...
1
vote
1answer
19 views
Show that the product of the Jacobian and the inverse Jacobian is 1
I have seen the following fact in a textbook, but am having trouble proving it. If the Jacobian ("stretch factor" for change-of-variables) is given by
$\left | \frac{\partial (x,y)}{\partial (u,v)} \...
0
votes
0answers
9 views
Conditions to tell that the total flux in a vector field across a closed surface is zero
When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
2
votes
0answers
21 views
Points of Confusion About Second-Order Taylor Formula of Taylor's Theorem For Many Variables
My textbook has written the following for the second-order Taylor formula of Taylor's theorem for many variables:
$$f(\mathbf{x}_0 + \mathbf{h}) = f(\mathbf{x_0}) + \sum_{i = 1}^n h_i \dfrac{\...
-1
votes
0answers
9 views
Partial derivatives of a homogenous function
See demo of the theorem here : https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/hom/t)
This is actually a follow up question on this one :
Help to understand the proof of partial ...
0
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0answers
13 views
Regarding moving the differential inside of a double integral.
Forgive me it has been a few years since I took multivariable calculus so I am unsure about something.
Say I am given something like $\int_x^y \int_x^y h(a,b)f(a,b|x,y)dadb = g(x,y)$.
Specific ...
5
votes
1answer
60 views
Evaluate $\int\int\int_{[0,1]^3}\frac{dxdydz}{(1+x^2+y^2+z^2)^2}$
As in the title I have to evaluate this triple integral.
$$\int\int\int_{[0,1]^3}\frac{dxdydz}{(1+x^2+y^2+z^2)^2}$$
I've been trying to solve this since a week ago.
The first thing I've done was ...
0
votes
1answer
12 views
integrate over one cordinate like partial derivative
Probably my question is related to this question, but this question does not provide the answer to my question.
Let $f:X\times Y \rightarrow \mathbb{R}$. Suppose I am interested in integration of $f(...
-3
votes
0answers
27 views
0
votes
2answers
51 views
Proof inequality $\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$
I'm asked to prove the inequality
$$\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$$
After playing around for a while I was able to find (...
-1
votes
1answer
29 views
a question about gradient
Let $f(x,y)=\ln\|\mathbf r\|$ where $\mathbf r=x\mathbf i+y\mathbf j$. Show that $\nabla f=\frac{\mathbf r}{\|\mathbf r\|^2}$.
I attempt to calculate $\nabla\mathbf r$. But I have no idea how to ...
1
vote
2answers
40 views
How to find an area between bounds with polar coordinates? (Double integral)
The problem is: Find the area of the region that is bounded by x = 2 on the right and y = x/
√ 3
above, and by the circle $x^2$ + $y^2$ = x
I have a general idea of how to find the ...
4
votes
2answers
58 views
Differentials in Multivariable Calculus
Does the idea of composing/decomposing the fraction notation of the derivative from/into differentials apply in multivariable calculus? I realize that this practice is considered non-standard and ...
0
votes
1answer
31 views
$\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$
Find the integral $\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$
So I tried the following substitution:
$$u = xy$$ $$v = x-y$$
And want to find the ...
0
votes
1answer
44 views
Is the local max point also a global max?
Consider the following function $f$ of three variables, defined on $\mathbb{R^{3}}$: $$f(x,y,z) = 15x + xy - 4x^{2} - 2y^{2} - z^{2} + 2yz + 7$$
Is each of the critical points a local maximum? Is ...
1
vote
2answers
29 views
Temperature in a point is given by $T=xyz$. If you are in the point $(1,1,1)$, in which direction you should go to keep the same temperature?
I've found the following exercise:
Suppose that the temperature in a point is given by $T=xyz$. If you are in the point $(1,1,1)$, in which direction you should go to keep the same temperature?
I ...
0
votes
0answers
25 views
Existence of an isolated point such that $\frac{\partial u}{\partial y} = 0$.
Let $u$ be a smooth real valued function on $\mathbb{R}^2$. Is it possible for the derivative $\frac{\partial u}{\partial y}$ to have an isolated zero? I think the answer is no. Here is my reasoning:
...
0
votes
1answer
33 views
Is $z = x^2y^3(1-x-y)$ convex or concave?
Is there some kind of trick to defining the domain of the concavity/convexity (if it exists)? I have no idea how to work with the resultant hessian
-1
votes
1answer
13 views
Not sure about 2nd half of this question
I get the first part that you need the partial derivate of x in respect to y and y in respect to x to be equal for it to 'maybe' be a gradient. After this however I don't see what im trying to do or ...
1
vote
0answers
32 views
How to find the Volume of Dubai's Museum of the Future (An ellipsoid with an ellipsoidal hole in it)?
I'm an IB Mathematics HL student doing a math research report for school, I need to be able to model Dubai's Museum of the Future building. Do you know how I could model this shape mathematically and ...
0
votes
1answer
43 views
Finding an analytical solution to a simple 2D Finite Element Method problem
diagram
Is it possible to find the value of scalar function u(x, y) anywhere in the region $\Omega$, given the following:
$\nabla \cdot$ ($\nabla$u) = f,
u(x,y) = g when y = 0, L$_2$
($\nabla$u)$...
1
vote
0answers
22 views
Derivative with single integral of variable limits, but with separate integration variable.
Say I have the following:
$$f(t) = \int_0^t{e^{s^3\sin(t)}ds}$$
How do I compute $f'(t)$?
My first thought was to use FTC, but the $s$ variable makes this confusing. I'm not completely sure where ...
0
votes
1answer
30 views
How to show whether limit of this function exists or does not exist?
The function is
$$f(x,y)=\frac{2x}{x^2+x+y^2}$$
I want to check whether
$$\lim_{(x,y)\to(0,0)} f(x,y)$$
exists or does not exist. I tried in polar coordinates and get the function
$$f(r, \theta)=\frac{...
1
vote
0answers
29 views
Prove that it is possible to isolate $z$ as a function of $(x,y)$ in $x^2 z^2-z^3 y x=0$ near the point $(1,1,1)$
Prove that it is possible to isolate $z$ as a function of $(x,y)$ in $x^2 z^2-z^3 y x=0$ near the point $(1,1,1)$, but not near the origin.
I just need the idea for start, cause its supposed to be a ...
0
votes
0answers
23 views
Validity of axes renaming
The position vector is $\vec{r}=(x,y,z)$.
$\vec{A}$ is a vector field, $\vec{A}=(a_1,a_2,a_3)$
and $C$ is a constant.
We have the function
$$f=f(a_1x,a_2x,a_3x,Cx^2).$$
Can one then say, that ...
1
vote
2answers
22 views
Existence of a double integral of given function
Problem
Let f be defined on the rectangle Q=[0,1]×[0,1] . f(x,y) is 1 when x=y and 0 elsewhere.
Prove the double integral exist and equal to zero.
Double
I have no idea how to prove the existence ...
0
votes
1answer
30 views
Relation between Riemann sums and oscillation of a bounded function
Let $I$ denote a closed (hyper)rectangle in $R^n$ and $f:I \rightarrow \mathbb{R}$ be a bounded function. We use the following notations.
(1) $\nu(f;a)$ is the oscillation of the function $f$ at a ...
4
votes
1answer
117 views
Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$
I've got a problem calculating the volume of the top half of a pseudosphere.
The pseudosphere is parametrised by
$$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
0
votes
1answer
11 views
Show equivalence of minimization techniques using Lagrangian multiplier.
Define a cost function :
$$L(x,w) = \frac{1}{2} \sum_{i=1}^N\left(y^{(i)} - w^T\phi\left(x^{(i)}\right)\right)^2 + \frac{\lambda}{2}\sum_{j=1}^M w_j^2$$
Show that minimizing $L$ is equivalent to ...
0
votes
1answer
13 views
Showing how the Jacobian connects volumes for change of coordinates
I'm having trouble understanding how exactly the Jacobian is transforming volumes.
To understand this in more detail... if I have a two-dimensional integral and I'm trying to change the integration ...
0
votes
0answers
43 views
Partitions of unity of a compact subset of a smooth manifold
Let $M$ be a smooth manifold (without boundary if you want). Let $K$ be a compact subset of $M$. Let $\{(U_\alpha,\phi_\alpha)\}_{\alpha=1}^s$ be a finite number of charts of $M$ such that $\{U_\alpha\...
3
votes
0answers
53 views
A function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ that maps a circle to a circle
Suppose f is a continuous function from $\mathbb{R}^2$ to $\mathbb{R}^2$ that maps a circle to a circle. How do I prove that f is differentiable? Will the function still be differentiable if ...
1
vote
0answers
12 views
Linear approximation of $3$ variable function and its maximum error?
I've got a problem which asks me to find linear approximation of multi-variable function and its maximum error.
Here's the problem :
By about how much will
$$g(x,y,z)=x+x\cos(z)-y\sin(z)+y$$
...
-1
votes
1answer
19 views
Find limits along different paths of the form $y = mx$ [on hold]
I need to show that the limit:
$\lim\limits_{(x,y)\to(0,0)}f_{y}(x,y)$ does not exist along different paths of the form $y = mx$, for $m$ is a real-number.
$$f_y(x,y) = \frac{-3y}{\sqrt{x^2 + 3y^2}...
2
votes
1answer
41 views
Poincare type inequalities
I want to prove if following inequality holds:
$$\int_0^1(f')^2\ dx\geq f^2(1)-f^2(0)$$
where $f$ is a function in $H^1([0,1])$ satisfying $\int_0^1f \ dx=0$. It is actually a one dimensional case of ...
0
votes
2answers
26 views
If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist?
Suppose you trace the path y=mx of a multivariable function f(x,y) to find the limit as (x,y)->(0,0). If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist?
...
0
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0answers
19 views
Derivative with respect to Vector: Notation
Let $x$ and $y$ be vectors in $\mathbb{R}^n$.
First, does the above notation imply that $x$ and $y$ are column vectors, or is that yet to be defined?
But to my main question: find
$$\quad \frac{\...
0
votes
1answer
22 views
Level curves of exponential
I have the following two-variable function:
f(x,y) = exp(-x^2-(y-1)^2)
And I need to compute/sketch the level curves for ...
0
votes
1answer
35 views
When a mathematician says $f(y,x)$ is strictly increasing in $x$, what do they mean?
when someone says $f(x,y)$ is increasing in $x$,
do they mean partial differentiation $\frac{\partial f(x,y)}{\partial x}$ or total differentiation $\frac{d f(x,y)}{d x}$?
If $y=h(x)$, then $f(x,y) = ...
0
votes
0answers
31 views
Another difficult 2D trigonometric integral
This is a follow-up question to A difficult 2d trigonometric integral. Unfortunately, I had a mistake in my calculations and I need a different (yet similar) seemingly simple integral solved:
$$\...
2
votes
1answer
53 views
differentiate $g(f(x),x)$ with respect to $f(x)$
Suppose I have a function $g(y,x)$ which is differentiable with respect to both arguments. I know that $y=f(x)$ is bijective, and thus the inverse function exists $x = f^{-1}(y)$. My question is when ...
0
votes
2answers
31 views
Prove that if $f(x, y)$ is continuous on $(0,0)$ then the function $g(x,y)=xyf (x,y)$ is differentiable on $(0,0)$
I made the following:
$$\lim_{(x,y)\rightarrow(0,0)} \frac{xy f(x,y) - 0 -g_{x}(0,0) - g_{y}(0,0) (y-0)}{((x-0)^2 + (y-0)^2)^{\frac{1}{2}}}$$ $$=\lim_{(x,y)\rightarrow(0,0)} \frac{xy f(x,y)}{(x^2+y^2)^...
1
vote
1answer
48 views
A difficult 2d trigonometric integral
I'm trying to solve the following seemingly simple integral - so far, without success:
$$\int_{a}^{b}\int_{a}^{y}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}$$
For some $0<a<b$....
2
votes
2answers
82 views
Why do we have to try all the paths in multivariable limits?
I don't get the idea of trying "all" paths to know the value of the limit. The way I see it is I have to "scan the area" around the point, and that would be enought. for example, if all the straight ...
0
votes
2answers
24 views
Questions on changing bounds of integration for double-integrals
I'm having difficulty understanding how to properly change these bounds of integration. When I set up the bounds, which x or y bound is supposed to depend on the other?
$$\int_{0}^{\pi/2}\int_{0}^{\...
0
votes
1answer
30 views
Proving limits existance
Im supposed to show and justify if the limit exists of this fucntion.
$$\lim_{(x,y)\to(0,0)}\frac{\cos x-1-x^2/2}{x^4+y^4}$$
One way the solution says is fine is to approach along the line $y=0$ which ...
0
votes
2answers
26 views
Limit of double integral at origin
I have a problem set with the following problem:
“Prove that, if $D_1={(x,y) | x^2+y^2 \leq r^2}$, then $\ lim_{r\to 0} \frac{1}{\pi r^2} \iint_{D_1} f(x,y) dA =f(0,0)$.
I am just stuck, as I ...
0
votes
0answers
44 views
Is $f(x,y, z)$ considered three or four dimensional?
I am doing path integrals video for both scalar fields and there is some confusion on my part how dimensions are treated.
$F(x,y) = z$ is considered $3$ dimensional in the video lecture as I see it....
0
votes
2answers
29 views
Proof convergence of vectors
I want to prove convergence of sequence.
Let $a_k$ be a sequence in $\mathbb{R}^k$ with $a_k$=$(a_{ki}...a_{kn})$ for all k element of natural numbers, and let $a=(a_1,...a_n)$ element of $\mathbb{R^...
0
votes
1answer
27 views
2nd derivative of f(x(t) , y(t)) [duplicate]
So the first derivative w.r.t. t:
$\frac{dz}{dt} = \frac{\partial f}{\partial x} \times \frac{dx}{dt} + \frac{\partial f}{\partial f} \times \frac{dy}{dt}$
How would I find the 2nd derivative?
...
