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).

learn more… | top users | synonyms

1
vote
1answer
13 views

Mass and center of mass using double integrals

Disclaimer: This was given as a homework from college but the teacher didn't teach us anything about density or mass or anything related. A lamina has the form of the region limited by the parabola $ ...
-2
votes
0answers
10 views

Find the average value of $f(x; y) = y$ over the region $ D$.

Consider the region $D$ in the $xy$-plane bounded by $y = -1, x = -1, x = y - y^3$ and $x = y^{1/2}- 1$. Find the average value of $f(x; y) = y$ over the region $D$. i don't know what to do with ...
0
votes
1answer
9 views

Existence of a Non-Linear Function Satisfying Certain Conditions

Let $f: \mathbb{R} \rightarrow \mathbb{R^m}$ and $N: \mathbb{R} \rightarrow \mathbb{R^m}$ be two mappings satisfying: $$\lim_{h\to0} \frac{f(a+h)-f(a)-N(h)}{h}=\vec 0.$$ If $f'(a)$ exists and is ...
0
votes
2answers
20 views

Evaluate $\int\int_B (x+y)dxdy$, where B is the rectangle in the xy plane with vertices (0,1), (1,0), (3,4)and (4,3).

So I found the linear equations of the sides of the rectangle: $y=-x+1, y=x-1, y=-x+7, y=x+1$ How do I use this information to construct a transformation $T(x(u,v), y(u,v))$ for a simple rectangle ...
0
votes
2answers
28 views

Matrix Calculus and Matrix Derivatives

Consider a map $f : \mathbb R^{n\times m} \to \mathbb R^{p \times l}$ between matrix spaces, what is the differential of such a mapping? I looked at a really simple example, $\operatorname{id} : ...
2
votes
0answers
21 views

Stokes Theorem Manifold with Corners Proof

I'm working through the proof for Stokes' Generalized Theorem for Manifolds and have a questions about corners. I've seen several proofs for manifolds with corners by creating diffeomorphisms to ...
-1
votes
1answer
19 views

for what value of $d$, $ S1: z=x^2+y^2 + d $ and $ S2: x^2+y^2=z^2 $ surfaces are tangent? [on hold]

for what value of $d \in R$(if any) the 2 surfaces are tangent ? $ S1: z=x^2+y^2 + d $ $ S2: x^2+y^2=z^2 $
2
votes
1answer
10 views

Finding the flux of the surface $z=a-x^2-y^2$ lying above $z=b<a$

I am stuck on the following question about finding the flux: Find the flux of $\vec F= (x,y,z)$ upward through the part of the surface $z=a-x^2-y^2$ lying above the plane $z=b<a$ My attempt: ...
1
vote
2answers
24 views

Evaluating a seemingly simple limit, using continuity of the partial derivative

I am stuck with a seemingly easy problem. I have a function $f(x,y)$ which has continuous first derivatives. Now, I want to show that: $$ \lim_{h \to 0}\dfrac{f(a+hu,b+hv)-f(a,b+hv)}{hu} =f_x(a,b)$$ ...
2
votes
2answers
21 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...
0
votes
0answers
25 views

Using Gauss divergence theorem on cylinder

Use Gauss’s divergence theorem to compute $$\iint \limits_S F ·n \, \, dS $$ where $n$ is the outward normal for the following: (a) $S$ is the exterior surface of the cylinder $x^2 +y^2 ≤ 1$, and $0 ...
0
votes
0answers
18 views

How can we maximize the following functional?

$\max_{} \; -\frac{1}{6} \lambda_1^2 + \lambda_1 + \int_0^1\left( \lambda_1 \lambda_2(t) (1-t) - 0.5 \lambda_2^2(t)- 2.5 \lambda_2(t)\right) dt$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
1
vote
1answer
15 views

Finding the volume of the following solid using triple integrals

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder $x^2 +y^2 =4$ and the plane $z+y=3$. I found the integral bounds just fine. So I have $\int_{0}^{2} ...
0
votes
0answers
11 views

Prove the surface integral of a constant vector field over a surface is equal to the area of the surface times the norm of vector field

I have a question as below. Prove that the surface integral of a constant vector field $F$ over a surface $S$ is equal to the area of $S$ times the norm of $F$, or find counter example. I think it ...
0
votes
2answers
20 views

Prove constant function using Gauss Theorem

Given $\vec F = \nabla f$ and f is a smooth function from $\mathbb{R}^3$ to $\mathbb{R}$. And, $$\iiint_B f(\nabla \cdot \vec F) \,dV = \iint_{\partial B} (f\vec F) \cdot d\vec S $$ where $B$ is any ...
0
votes
0answers
11 views

Show that $U = f_{e_\delta}$ in cylindrical coordinates, where f is a function you should identify

Say I have a function $$U(x,y,z) = \frac{ayi - axj}{x^2+y^2}; a>0 \text{(constant)}$$ and I'm required to find in Cylindrical Coordinates $$U=F_{e_\theta}$$ How would I go about doing this?
1
vote
1answer
12 views

Find the divergence of $U(x,y,z) = \frac{ayi-axj}{x^2+y^2}; a\gt0$

how would I find the divergence of the function $$U(x,y,z) = \frac{ayi-axj}{x^2+y^2}; a\gt0$$ would it just be this? $$ \nabla \times F = \frac{\partial F_x}{\partial x}+\frac{\partial ...
0
votes
0answers
26 views

how to prove torus as a 2 dimensional manifold

Consider equation of torus in 3 dimension $(R-\sqrt{(x^2+y^2)})^2+z^2 = r^2$ where $R$ is larger radius and $r$ is smaller radius. how to prove that it is 2 dimensional manifold? I tried ...
3
votes
1answer
29 views

Spivak Calculus on Manifolds: Problem 2-13

I'm going through Spivak's Calculus on Manifolds, and I'm currently working on Problem 2-13 part (b). The problem statement is If $f,g: \mathbb{R} \rightarrow \mathbb{R}^{n}$ are differentiable ...
0
votes
0answers
25 views

Surface Integral, Stokes Theorem, Divergence theorem

We were given problems to work out before asked the meaning, but I have done them and am pretty confident with the answers, however I do not know the meaning of what I calculated. Please let me know ...
2
votes
0answers
17 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
1
vote
1answer
18 views

What is the Hessian matrix of $x\mapsto f(Ax+b)$?

Let $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^n$ $f\in C^2(\mathbb{R}^n)$ and $\tilde{f}(x):=f(Ax+b)$ for $x\in\mathbb{R}^n$ It's easy to prove that $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
3
votes
3answers
45 views

What is the difference between a scalar and a vector field?

Could someone please indicate precisely the difference between a scalar and a vector field? I find no matter how many times I try to understand, but I always am confused in the end. So what exactly ...
2
votes
0answers
32 views

Application of the implicit function theorem on an abstractly defined function

I have worked with Thomas C.T. Michaels Analysis II to study the application of the implicit function theorem and I have a pretty solid idea of how to apply it to multivariate functions but I cannot ...
0
votes
0answers
11 views

Sufficient condition for constrained extrema

When study Lagrange multiflier theorem. I try to get a sufficient conditions for constrained extremum due to the statement of Lagrange multiflier theorem, that is Given p+1 countinously ...
0
votes
1answer
35 views

Double integral (choice of) change of variables

I'm looking for a way calculate the following integral: $$\iint_D\frac{(x-y)^2(1+2y)}{(1+x+y^2} d(x,y)$$ With $D=\{(x,y)\in \mathbb{R}^2 : 0 \leq x+y^2 \leq 4 \mbox{ and } x\leq y\leq x+2\}$. what ...
1
vote
2answers
28 views

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$ where $k(x,y) = \begin{cases} y(x-1), & \text{ $0\leq y<x\leq 1$} \\[2ex] x(y-1), & ...
-1
votes
1answer
29 views

How to prove that extreme values don't exist [on hold]

Let $f(x,y)=(1-2x^2-y^2)xy$, how do I prove that the $f$ does not admit extreme absolutes?
1
vote
0answers
26 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
2
votes
3answers
74 views

How to create very hard problems on Lagrange Multipliers

This is a rather odd request. I only recently started studying the Lagrange Multipliers, and was given a task to create some challenging (as much as possible) problems on them and also provide ...
1
vote
1answer
41 views

Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb ...
2
votes
1answer
44 views

Find a rigorous reference that prove the following integration by parts formula in higher dimension?

My professor in the real analysis class had state the following in class but forgot to put the reference of this formula in the power point slide. The formula for integration by parts can be ...
2
votes
1answer
37 views

Where did I got wrong with this surface integral

It appears that I don't quite have surface integrals like I thought I did. The following is a problem from the back of the book (not homework because it wasn't prescribed but I'm working it to ...
0
votes
0answers
23 views

Differential of Gibbs free energy (dG) in terms of dT and dV [on hold]

The heat capacity, isothermal compressibility, and thermal expansion coefficient can be in the answer. A hint is to use the upstairs-downstairs-inside-out rule to help.
0
votes
1answer
25 views

How do I parametrise this line integral properly?

$ F (x,y,z) = ( zy + sinx , zx - 2y , yx-z ) $ is the vector field. Find the line integral of F which has curve C given as $ x = y = z^2 $ between (0,0,0) and (1,1,1) I first did this: Take $ z=t ...
2
votes
0answers
36 views

Exercise 2.36 in 'Vector Calculus, Linear Algebra and Differential Forms' (Hubbard)

Let $A$ be an n by n diagonal matrix with diagonal entries $\lambda_1$ to $\lambda_n$, and suppose that one of the diagonal entries, say $\lambda_k$, satisfies $inf_{k\neq j}|\lambda_k - \lambda_j| ...
3
votes
1answer
31 views

Analogs to vectors — *unoriented* line segments

A real vector can be thought of as an oriented line segment. Linear algebra and multivariable calculus can be taken pretty far just by considering these types of objects (obviously there are ...
8
votes
2answers
32 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
0
votes
1answer
17 views

convert from spherical co-ordinate system to cartesian

I want to convert $ρ=\cos{ϕ}$ to Cartesian system , after conversion my answer is $z=x^2+y^2+z^2$ , but its not a sphere , what have I done wrong?
0
votes
0answers
27 views

Which solution is correct?

This is a direct application of Stokes Theorem. So $\int_\Omega div F\; dv=\int_ {\partial \Omega} F.n \; dS$ $\Rightarrow 30=\int_S F.n \; dS+\int_D F.n \; dS\Rightarrow \int_S F.n ...
0
votes
0answers
28 views

Understanding notation - strange use of the del operator

I'm currently reading a paper with the following notation with the del operator which i have never encountered before: Does $\nabla _m$ just mean $\frac{\delta}{\delta \mathbf m} $ ? Furthermore, I ...
1
vote
0answers
18 views

I am trying to find the component of b in the direction perpendicular to a. I am trying to find an alternative route to this problem. Does this work?

c being the component of b in the direction perpendicular to a. So I used the triangular law regarding vectors. I wish I could draw a picture to make it more clear. But ill try to explain... proj ...
4
votes
2answers
32 views

Line Integrals FT usage on this strange vector field: so what are the exact conditions?

I really tried thousands of things before deciding to ask here. Searched all over the internet for an answer, but failed to find it. Let's get started with the Fundamental Theorem of Line Integrals. ...
2
votes
1answer
27 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,t) | \leq \frac ...
1
vote
1answer
17 views

Having Trouble finding a simplified power series representation.

Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
3
votes
1answer
32 views

Find the gradient of $f^*(x)=\langle (\nabla f)^{-1}(x),x)\rangle-f( (\nabla f)^{-1}(x))$ for $x \in \mathbb{R^n}$

I am stuck at the following exercise which serves as a preparation for the upcoming exam: Let $U \subset \mathbb{R}^n$ be open and $f \in C^2(U, \mathbb{R})$ such that $\det Hf(x) \neq 0, \forall ...
1
vote
2answers
23 views

Multivariable calculus converting from cartesian coordinates to cylindrical coordinates

How do I convert this from cartesian coordinates to cylindrical coordinates? I am really confused. $$\int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{x^2+y^2}^4 (x) \ dz \ dy\ dx$$ I changed ...
4
votes
2answers
21 views

Conceptual explanation of integral of divergence.

$\textbf{My understanding of divergence:}$ Consider any vector field $\textbf{u}$, then $\operatorname{div}(u) = \nabla \cdot u$. More conceptually, if I place an arbitrarily small sphere around any ...
1
vote
0answers
23 views

Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
0
votes
1answer
26 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...