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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3
votes
2answers
24 views

Volume of Region Paraboloids

How do I find the volume of the solid region which is bounded by $z=2x^2+2y^2$ and $z=3-x^2-y^2$? So I first realized that these two functions are paraboloids and I have to find the volume of their ...
1
vote
0answers
25 views

Why do we take the dot product with the normal vector when we do Stokes' Theorem?

So this part I'm struggling with on Stokes' Theorem: $$\iint_S ~(\text{curl}~\vec{F} \cdot \hat{n})~ dS$$ I don't really understand why we would want to dot it with the unit normal vector at that ...
1
vote
2answers
62 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial ...
2
votes
1answer
20 views

Properties of Curl.

Could anyone please let me know which of these statements is true. 1)$$\text{curl}~{\vec{F}}=0 \implies \vec{F} ~\text{is conservative.}$$ 2) $$\text{curl}~{\vec{F}}=0 \impliedby \vec{F} ~\text{is ...
0
votes
1answer
12 views

calculate similiar volumes, but with little diffrence

There are to pretty similiar volumes, but there is something in the calculation, that I dont understand clearly and I would be glad if someone help me. The first (Vivani volume): a volume that ...
1
vote
1answer
28 views

Question about continuity of piecewise function of two variables

Let $$ f(x,y)= \left\{ \begin{array}{ll} \left(x\sin\left(\frac{y}{x}\right),\frac{\cos (y) -1}{y}\right) & x \neq 0 \wedge y \neq 0 \\ (0,0) & x = 0 \vee y = 0 \\ \end{array} ...
0
votes
0answers
6 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...
0
votes
1answer
14 views

Extrema function of two variables problem

A rectangular box with a square base is to be constructed from material that costs 9 dollars per $ft^2$ for the bottom, 7 dollars per $ft^2$ for the top, and 4 dollars per $ft^2$ for the sides. Find ...
1
vote
3answers
61 views

Vector space or vector field?

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field ...
0
votes
1answer
14 views

Line integral over a intersection of a cylinder and a plane.

Here is the question I'm trying to answer: Compute $$\int_C(y−z)\, dx+(z−x)\, dy+(x−y)\, dz$$ where $C$ is the intersection of the cylinder $x^2+y^2 = 1$ and the plane $x−z = 1$. I don't ...
0
votes
2answers
23 views

Moment of inertia about the origin of an ellipsoid?

Find the moment of inertia about the origin of an ellipsoid. Heres what I did but I believe it is incorrect: $$I_o= \iiint_{V_e}{(x^2 +y^2 +z^2)\rho dx dy dz} $$ Making Substitution of $aX=x \ bY=y \ ...
1
vote
2answers
19 views

Harmonic function and surface integrals.

I am trying to give a solution to this question but I'm getting stuck. Let $f$ be an harmonic function $V \to \mathbb R$ where $V$ is a solid in $\mathbb R^3$ bounded by a solid surface $S$ with ...
0
votes
0answers
26 views

Geometric interperation of line integral - example

I have hard times figuring out geometric interpretation of line integrals. Here is one example from my book: Calculate area of cylinder $x^{2}+y^{2}=ax$ sliced with sphere $x^{2}+y^{2}+z^{2}=a^{2}$. ...
0
votes
1answer
42 views

Books on multivariable calculus

I'm looking for a book that covers the following subjects: multivariable functions, extremes of multivariable functions, integration, implicit function theorems, functions defined by integrals, vector ...
1
vote
4answers
38 views

How to find $\frac{\partial^2u}{\partial x \partial y}$ given $u=\sin(x\sin^{-1}y)$?

How to find $$\frac{\partial^2u}{\partial x \partial y}$$ given $$u=\sin(x\sin^{-1}(y))$$? I have calculated $$\frac{\partial u}{\partial x }=\sin^{-1}(y)\cdot \cos(x\sin^{-1}(y))$$ but get stuck ...
0
votes
1answer
25 views

What exactly does changing the variables in partial derivatives mean?

From differential geometry I have learned that: $$ \partial_{x + y} = \partial_x + \partial_y $$ Now trying to prove this property for partial derivatives as I know them from multivariable calculus ...
1
vote
0answers
49 views

The maximum of a functional

Is the following statement true or false? $$ \max F \left( \theta \right)= \int_{\rho_{min}}^{\rho_{max}} g \left( \rho \right)\pi\left(\rho,\widehat{\theta \left( \rho \right)} \right)d\rho$$ ...
0
votes
1answer
46 views

Hessian of composite function

Let $\cal{J}(y) : \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore, suppose $y := h(x)$ with the nonlinear mapping $h : \mathbb{R}^n \rightarrow \mathbb{R}^n$ twice differentiable. The Jacobian of ...
0
votes
3answers
25 views

Extrema Function of $2$ variables [on hold]

I do not know how to set this problem up. Any insight as to how to get the equation would be great. It is John's birthday and his parents want to make him a cake in the shape of a rectangular box. ...
3
votes
1answer
45 views

Is this function in $L^1_{loc}(\mathbb R^3)$

It seems such a trivial question, but for whatever reason I don't understand. Let $u: \mathbb R^3 \to \mathbb R $ be $$u(x) = \frac 1{4\pi |x|}$$ The book says that $u \in L^1_{loc}(\mathbb R^3)$ ...
0
votes
1answer
29 views

Interpreting the formula of the derivative of a multivariable function.

Let $f:R^2\to R $ be a differentiable function. Then we know that $$f (x+h, y+k)-f (x, y)=\frac {\mathrm{d}f }{\mathrm{d}x }\mathrm{d}h+ \frac {\mathrm{d}f }{\mathrm{d}y }\mathrm{d}k.$$ The way I ...
1
vote
1answer
9 views

Flux integral/notation of position vector

"Find the flux of $\mathbb{F} = \frac{m\vec{r}}{|\vec{r^3}|}$ out of the surface of the cube $0\leq x,y,z\leq a$ " Two questions regarding this: When I first "solved" it, I considered $\vec{r}$ the ...
1
vote
0answers
34 views

Derivative of the Kullback Leibler Divergence

If: $$ H (\pi(t)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log \frac{\pi_i(t)}{\mu_i(t+1)} $$ How do I interpret: $\nabla H(\pi(t) | \mu(t+1) )$? Would it be the vector: $$ \left ( \frac{\partial ...
0
votes
0answers
27 views

Is this idea correct?

Given a curve passing through point $(p_0, v_0)$ and defined in a standard way as $k(p, v) = k(p_0, v_0)$, i can find the 1st term,a 2nd term b and so on by expand k in taylor series and consider ...
1
vote
1answer
39 views

Using triple integrals to find volume

Find the volume of the solid defined by the inequalities $0 \le z \le y \le x \le 1$. I know I have to use triple integrals to solve this problem, but I am pretty confused as to how I should ...
0
votes
0answers
13 views

Finding Extrema of a Function Restrained by a Parametrized Surface

Where on the parametrized surface $r(u,v)=⟨u^2,v^3,uv⟩$ is the temperature $T(x,y,z)=12x+y−12z$ minimal? Find all local maxima, local minima or saddle points. I know that one has to insert $r(u,v)$ ...
0
votes
0answers
24 views

Change of Variable in Multidimensional Integral

I'm trying to understand part of the paper published here: http://www-stat.wharton.upenn.edu/~tcai/paper/Testing-Covariance-Matrix.pdf. I'll reproduce the relevant portions here so hopefully you will ...
0
votes
3answers
63 views

Continuity but not differentiability of $f(x,y)$ at $(0,0)$

Define $f(x,y) = 0$ if $(x,y) = (0,0)$ and $f(x,y) = \dfrac{x y^2}{x^2+y^2}$ if $(x,y) \neq (0,0)$. a) Show that $f$ is continuous at the origin. b) Prove $(D_uf)(x)$ exists at $(0,0)$ ...
-3
votes
1answer
61 views

Write integral in different order [duplicate]

How can I rewrite, the following triple integrals into different orders?: $$\int^1_0\int^{z^2}_0\int^y_0{f} \, \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ In the orders $\mathrm{d}z \, \mathrm{d}x \, ...
1
vote
1answer
53 views

Rewriting Triple Integral in different order

How can I rewrite, say: $$\int^1_0\int^{z^2}_0\int^y_0{f}dxdydz$$ In the orders $dzdxdy$ and $dydzdx$? Would I have to solve this triple integral all the way or is there a simple way to do this?
2
votes
2answers
56 views

Need help with Double Definite Integration

I need help in solving this double integral: $$\int\limits^2_{-2}\;\;\int\limits^\sqrt{4-x^2}_{-\sqrt{4-x^2}}{(x^2+y^2)^{5/2}}dy\,dx$$ Maybe introducing polar coordinates might help?
2
votes
1answer
76 views

Volume of Solid Defined by Inequalities

How can I find the volume of a solid defined only by inequalities? For example, in this case I have: $$0\le z \le y \le x \le 1$$ Can someone please explain to me step-by-step on how I can do this. ...
1
vote
1answer
54 views

Finding Surface area of Paraboloid [duplicate]

I am having trouble finding the surface area of the part of the paraboloid that lies in the first octant of $z=5-x^2-y^2$. So, I realized that the first octant refers to when, $x,y,z \ge 0$. How do I ...
0
votes
1answer
17 views

Use lagrange multultipliers to find the indicated extrema

maximize $f(x,y,z)=x+y+z$ subject to $x^2+y^2+z^2=1$ I do not understand this at all or where to go from here would appreciate some insight
0
votes
1answer
33 views

Find the partial deriavtive with respect to x and y

$$f(x,y)=\ln \frac{x-y}{(x+y)^2}$$ Use log properties I started with this $$\ln(x-y)-2\ln(x+y)$$ I got this for $x$: $$\frac{1}{x-y}-\frac{2}{x-y}$$ I got this for $y$: ...
-1
votes
3answers
50 views

How to generalize $f = \frac{u}{v}$, $u,v$ scalars, for vectors? [closed]

Suppose $f = \frac{u}{v}$ for some scalars $u,v$. How does one go about generalizing this for vectors $\mathbf{u,v}$? I think there is no concept such as division by a vector ..
1
vote
0answers
60 views

When can I calculate a derivative in a point?

Okay the title makes no sense. I have a two variable function, $f(x,t)$. When is it that $$ \left(\frac \partial{\partial x} f(x,t) \right)\bigg| _{t=0} = \frac{d}{dx} f(x,0)$$? My guess is that it ...
1
vote
1answer
35 views

Difference between a Fréchet derivative and a total derivative

I've heard many times that they are somehow similar and in some cases mean the same thing. Consider this function: $$f(x,y)=x^2y$$ I have to calculate the Fréchet derivative $f'(x_0,y_0)$ and some ...
1
vote
1answer
37 views

Diffeomorphism which has a zero

Let $f:B(x_0,r) \subset \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism between $B(x_0,r)$ and its image. If $|f'(x)^{-1}| \leq M$ for all $x \in B(x_0,r)$ and $|f(x_0)|<r/M$, show that ...
2
votes
2answers
164 views

Does the concept of a derivative a rate of change work for n dimensions?

I am trying to understand what exactly a derivative is. I understand the total derivative is a linear map. But I don't understand what happens to the idea of a rate. In high school calc, one is ...
0
votes
1answer
20 views

computing flux and circulation using Green's theorem

I must compute the outward flux and counter clockwise circulation of $F$ through and around $C$ using Green's theorem. $F=<xy,\:x+y>$, ...
0
votes
1answer
28 views

Equilibrium Points for 8th Degree Polynomial

I have an 8th degree polynomial that I need the zeros for. Is there even a way to explicitly solve one? Its for a diff equations review. I need to sketch the phase line, which is a breeze once I get ...
3
votes
1answer
173 views

How can I solve $\lim_{(x,y) \rightarrow (0,0)} \frac{xy\sin(x+y)}{x^2+y^2+|xy|}$?

How can I solve: $$\lim_{(x,y) \rightarrow (0,0)} \frac{xy\sin(x+y)}{x^2+y^2+|xy|}$$ I used Wolfram Alpha, but it said that it doesn’t exist. If I do it by myself I get: $$0 < ...
1
vote
0answers
48 views

What is the derivative of the gradient vectors with respect to a scalar field? [closed]

Let $\mathbf x$ be vectors and $\phi(\mathbf x)$ be a scalar field, does the following relation hold? $$\dfrac{\mathsf d}{\mathsf d\phi}(\nabla\phi(\mathbf x)) = \mathbf\bf 0$$ Here $\bf 0$ is a ...
0
votes
0answers
15 views

Asymptotic Curve of Time [closed]

Suppose you have a Hexadecimal String of M length, and you want to parse through that string seeking N smaller Hexadecimal Strings of K length. What on earth would this formula look like?
1
vote
0answers
33 views

A question about a system of PDE

It is well known that under suitable conditions, the symmetry of mixed second partial derivatives reads: $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}.$$ ...
0
votes
0answers
16 views

How can we visualise the direction of greatest increase of a function [closed]

How can we visualise the direction of greatest increase of a function (i.e. the gradient) is the vector sum of two perpendicular slopes in x and y directions i.e. $ \frac{\partial f}{\partial ...
0
votes
3answers
56 views

Can I argue that $g'$ is non zero in this case?

Consider two smooth maps $g,f$ given by $$ {\partial \over \partial x} g(x)= g'(x) = \int_0^1 {\partial \over \partial x} f'(u + t(x-u)) dt = \int_0^1 f''(u + t(x-u)) \cdot t dt $$ where $f' = ...
2
votes
1answer
33 views

Iterated Integral and Sign Change in Answer

Given the iterated integral $\int_0^1\int_x^{2-x}(x^2-y) \, dy \, dx$, the value for the type I integral is, \begin{align*} & \int_0^1\int_x^{2-x}(x^2-y)\,dy\,dx \\ = {} & \int_0^1 ...
0
votes
0answers
41 views

Change of variable and partial derivative

Let us suppose we consider the following change of variables $(t,r)\rightarrow (T,R)$ with $$ f(t,r) \frac{\partial}{\partial t}= \frac{\partial}{\partial T} \quad(1)\\ g(t,r) \frac{\partial}{\partial ...