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).
1
vote
1answer
25 views
Geometric Meaning of the Partial Derivative
Could anyone explain the geometric meaning behind the partial derivative?
I know that for a "normal" derivative, its geometric meaning is the slope of the tangent of a curve. From what I understand ...
-1
votes
1answer
26 views
derivative of $f\left(x_1\left(t\right),x_2\left(t\right)\right)$
What is the derivative of following function $f\left(x_1\left(t\right),x_2\left(t\right)\right)$, where $x_1\left(t\right)$ and $x_2\left(t\right)$ are dependent on $t$?
1
vote
2answers
19 views
Manipulating Vector Identity
I would like to expand then simplify (if possible) the following quantity.
$\nabla (a \cdot (C\, a))$
Where $a = a(x)$ is a vector valued function of $x$ and $C$ is a constant matrix.
0
votes
2answers
19 views
How should one think to get the radius of the resulting curve?
For example, the curve C is given as the intersection between
$$
C: x²+y²+z²=1, x+y+z=0
$$
Radius: 1
Another one:
$$
C: x²+y²+z²=1, x+y=1
$$
Radius:
$$
\frac{1}{\sqrt{2}}
$$
How should I think to ...
1
vote
3answers
34 views
Double Integrals. Simple question, don't understand their wording.
Make a sketch of the region over which
$$\int_0^{\pi/2} dx \int_0^{\sin(x)} dy$$
Would this be the same as
$$\int_0^{\pi/2} 1 dx \int_0^{\sin(x)} 1 dy$$
Which simply evaluates to ...
2
votes
5answers
55 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
1
vote
2answers
25 views
Necessary condition for local minima; non-negative Hessian matrix [duplicate]
The problem I have is the following. Any results on Taylor expansions etc. can be assumed:
Let F : R^n -> R be a C2 function. Let x_0 be a local minimum of F. Prove that the Hessian matrix of F is ...
11
votes
2answers
90 views
$\int_{\mathbb{R}}|f(t)|^2dt=\int_{\mathbb{R}}|f'(t)|^2dt$ implies $f(t)=\mathbb{x}_{i}|f(t)|$
Let $f \in C^{1}(\mathbb{R},\mathbb{R}^m)$ be such that $f$ and $f'$ are square integrable and
$$\{t:f(t)=0\} \subset \{t:f'(t)=0\}$$
$$ |\{t:f(t)=0\}|=n\in \mathbb{N}$$
Prove that if ...
1
vote
0answers
37 views
What is the value of $\int_{0}^{a} \int_{0}^{x} \frac{x}{y} \cosh{y} \; dy \, dx$?
I had the following double integral in my recent math examination:
$$\int_{0}^{a} \int_{0}^{x} \frac{x}{y} \cosh{y} \; dy \, dx$$
where $$a \in \mathbb{R}$$
I tried changing the order of the ...
0
votes
3answers
253 views
Prove $ \oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V $
I need help proving the following vector calculus identity:
$$
\oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V
$$
the ...
1
vote
0answers
31 views
Forward Time Centered Space Scheme on unit square, Stability Analysis
I'm stumped on the following problem:
Show that
$u(x,y,t)=\exp(1.68t)\sin(1.2(x-y))\cosh(x+2y)$ solves
$\frac{\partial u}{\partial t}-2\frac{\partial ^2 u}{\partial x^2}-\frac{\partial ^2 ...
3
votes
1answer
125 views
Does multivariate integral make sense and can it be evaluated?
I have very hard instructor for multivariate calculus. He ask if the next integral is well-defined.
$$ \iint\limits_D\ {\cos(z)\sin^3(z)\cos(y)\sin(y)\over ...
1
vote
1answer
51 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
1
vote
1answer
47 views
Line Integral of Every Positively Oriented Simple Closed Path - Green's Theorem
This question is from Example #5, Section 16.4 on P1059 of Calculus, 6th Ed, by James Stewart.
Given Question: If $\mathbf{F}(x,y) = \left(\dfrac{-y}{x^2 + y^2}, \dfrac{x}{x^2 + y^2}\right)$, show ...
0
votes
0answers
34 views
Euler's theorem for homogeneous functions
Let $\textbf{R}_{+}$ be the set of positive real numbers.
The following is a well-known theorem due to Euler:
A differentiable function $f:\textbf{R}^n_{+} \rightarrow \textbf{R}_{+}$ is positively ...
1
vote
2answers
48 views
Being inside or outside of an ellipse
Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside
$E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$
if some line passing trough $A$ ...
2
votes
3answers
96 views
Vector calculus for ellipse in polar coordinates
I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =)
I know that for a ellipse the parametric is $x=a\sin t$ , $b= b \cos t$, ...
1
vote
1answer
46 views
Circulation and line integrals.
The following is the problem I'm working on.
If $\overrightarrow {F} = <x,y^2z,-xy^2>$, calculate the circulation of $\overrightarrow {F}$ over the surface $z=x^2+y^2$ bounded by $C$ using a ...
0
votes
0answers
38 views
Consider the function $f(x,y)=2 \cos(x^2 +y^2) +xy \sin (x+y)^2 - x^2y^2 $
Consider the function $f(x,y)=2 \cos(x^2 +y^2) +xy \sin (x+y)^2 - x^2y^2 $
Determine the fourth order Taylor of $f$ in $(0,0)$. And use the inequality: $x^4 + y^4 + x^2y^2 - yx^3 - xy^3>0$ when ...
1
vote
1answer
43 views
Can Green's theorem be used in a plane other than the xy-plane?
In the following 2D case, Green's theorem solves the following problem:
$$\vec{F}=\langle{xy+\ln{(\sin{e^{x})},x^2+e^{y^2}}}\rangle$$
$$\oint_C\vec{F}\cdot{d\vec{r}}=\iint_Dx\space{dA}$$
where C is ...
0
votes
2answers
40 views
Vector Line Integral Question
I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.
Once again, I'm ...
1
vote
2answers
58 views
Volume of a set in phase space. How many dimensions?
Suppose I have a $6N$ dimensional space with points looking like this:
$$(r_x^{(1)},r_y^{(1)},r_z^{(1)}, p_x^{(1)}, p_y^{(1)}, p_z^{(1)},...,r_x^{(N)},r_y^{(N)},r_z^{(N)}, p_x^{(N)}, p_y^{(N)}, ...
0
votes
3answers
137 views
Differentiability, function of two variables
How can I prove that $f(x,y)=\mathrm{e}^{xy}$ is differentiable on $\mathbb{R}^2$. (Only by the following definition)
$\textbf{DEFINITION}$: Suppose that for every point $(a+\Delta x, b+\Delta y)$ ...
0
votes
1answer
33 views
Work to provide explanation on the definition of the area of a Jordan-measurable set
The problem is as follows:
Given this theorem:
Let $D$ be bounded & Jordan-measurable set
Let $f$ be a bounded function on $D$
And $f$ is continuous except for a set of zero ...
2
votes
1answer
1k views
Gradient of a vector field?
What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field... thank you.
-1
votes
0answers
63 views
What is $\lim\limits_{|a| \to \infty} \int_0^1 (G(x) - a_0 - a_1x)^2\,dx$?
Assume the integral of g from 0 to 1 is a finite #.
$$\lim_{|a| \to \infty} f(a) = \lim_{|a| \to \infty}\int_0^1 (G(x) - a_0 - a_1x)^2\,dx$$
$a= [a_0, a_1]$, as $|a| \to \infty$, we have $a_0^2 + ...
3
votes
1answer
62 views
Linearization of an implicitly defined function.
Problem:
Given the equation: $xz^{2}+y^{2}z^{5}=19$
Also given: (3,4,1) is a solution to the equation. This point is not the only solution.
1) Find dz/dx and dz/dy (through implicit ...
2
votes
2answers
54 views
Finding the range of a vector valued function
For a single valued function, I can infer if the function is monotone from its derivative.
For a vector valued function, is it possible to infer monotonicity from the directional derivative?
For ...
3
votes
1answer
24 views
Any arbitrary closed smooth curve bounds a orientable surface?
I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ?
I have no ...
4
votes
2answers
74 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
vote
1answer
33 views
Integration with change of variables (multivariable).
The following are the problems that I have been working on. It involves change in variables with 2,3 variables respectively.
(1)Let $R$ be the trapezoid with vertices at $(0,1),(1,0),(0,2)$ and ...
3
votes
1answer
57 views
Is the function identically zero?
Let $f(x, y)$ be a continuous, real-valued function on $\mathbb{R}^2$.
Suppose that, for every rectangular region $R$ of area 1, the double integral of $f(x, y)$
over $R$ equals 0. Must $f(x, y)$ be ...
0
votes
0answers
16 views
Finding rate of maximum temperature increase along surface
So I know that the rate of maximum increase of some function (say, $f(x,y)$) is given by the gradient ($\nabla f$), where the direction is the direction of maximum increase of the function, and the ...
1
vote
1answer
67 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
1
vote
0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
0
votes
1answer
52 views
Liouville's formula
I have some questions concerning a proof of Liouville's formula:
$$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE.
If the vectors in the columns of the fundamental matrix ...
0
votes
0answers
19 views
Prolate spheroidal coordinates
If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$.
$$\varphi (\alpha,\beta,\theta) =(
\sinh(\alpha)\sin(\beta)\cos(\theta),
\sinh(\alpha)\sin(\beta)\sin(\theta),
...
1
vote
1answer
174 views
Unit Tangent and Unit Normal Vectors — Calculus III Question
Consider the following vector function.
$$r(t) = \left\langle 2t \cdot \sqrt{2}, e^{2t}, e^{-2t}\right\rangle$$
(a) Find the unit tangent and unit normal vectors $T(t)$ and $N(t)$.
$T(t) =$
$N(t) =$ ...
0
votes
0answers
12 views
A general formula for the partial derivatives of $\sigma(\xi_1,\ldots,\xi_j,-(\xi_1+\cdots+\xi_m),\xi_{j+1},\ldots,\xi_{m})$.
Let $\sigma$ be defined on $(\mathbf{R^n})^m\backslash \{0\}$ and suppose it is adequately differentiable (that is, we can take as many derivatives as required to show this next statement). If ...
0
votes
1answer
17 views
prove that a function is a diffeomorphism
anyone can help me to prove that $f(\alpha,x,y,z) = (\sinh(\alpha) x, \sinh(\alpha)y,\cosh(\alpha)z)$ is a diffeomorphism? In fact, i'm not sure if it's a diffeomorphism.
1
vote
1answer
33 views
If $f:U\to \mathbb{R}$ is continuous and $(x^2+y^4)f(x,y) + (f(x,y))^3=1$, then $f$ is $C^\infty$
Let $f:U\to \mathbb{R}$ be continuous in $U \subset\mathbb{R}^2$, such that
$$(x^2+y^4)f(x,y) + (f(x,y))^3=1$$
for all $(x,y) \in U$. Prove that $f\in C^{\infty}$.
I'm learning the implicit ...
4
votes
1answer
69 views
Find the volume of the region bounded by $z = x^2 + y^2$ and $z = 10 - x^2 - 2y^2$
So these are two paraboloids. My guess is I would want to find the intersection of these two which would be $2x^2 + 3y^2 = 10$ and construct a triple integral based on its projection. No idea how to ...
1
vote
2answers
38 views
Vector Field Generating Variation Along Curve
I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following.
Suppose ...
1
vote
1answer
142 views
Change of variables for triple integrals
Question: Consider the one-to-one transformation $(u; v;w) \to (x; y; z)$ defined by the equations $u = x + y + z; uv = y + z; uvw = z;$ which maps the unit cube $U$ defined by $0 \lt u \lt 1, 0 < ...
0
votes
1answer
18 views
computing the speed of decreasing of the temperature of a function that depends on the position on the space
Let's suppose that the temperature of a particle at a given position $(x,y,z)$ is given by $ T(x,y,z)=e^{-(x^2+2y^2+3z^2)}$ where $x,y,z$ let's consider that $x,y,z$ are given in meters=m$
If the ...
1
vote
1answer
51 views
Curvature and Torsion problem
Calculate the curvature and torsion of
$$x= e^t\sin(t),\quad y= e^t\cos(t),\quad z= e^t$$
I'm not sure if I am doing this correctly since I am getting quite complicated results.
But I understand ...
0
votes
0answers
41 views
Let $f,g :\mathbb{R}^n \to \mathbb{R}$, such that $g(x) = f(x) + (f(x))^5$. If $g \in C^k$ then $f \in C^k$.
Someone can help me on this question ?
Section on the implicit function theorem.
1
vote
2answers
76 views
Find the area of $A = \{ \langle x,y\rangle \in \mathbb{R}^2 \mathrel| (x+y)^4<a x^2 y,\ x>0 \}$?
I can't really think of how to set the limits
1
vote
1answer
13 views
Neumann problem, stuck on a boundary condition.
I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:
A solution is sought to the Neumann problem ...
1
vote
1answer
222 views
Parametric curve of intersection - line integral with respect to arc length
This comes from Apostol's Calculus, Vol. II, Section 10.9 #14:
A uniform wire has the shape of that portion of the curve of intersecion of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the ...
