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
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1answer
289 views

How to Handle Two-Center Bipolar Coordinates?

In my problem, I want to integrate a $2D$ function $f(x,y)$ which explicitly depends on the vector $ \vec{r}_1=\vec{r}-\vec{R}_1 $ and $\vec{r}_2=\vec{r}-\vec{R}_2$, where $\vec{R}_1=(a,0)$ and $\vec{...
1
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0answers
29 views

Proving that a function is coercive

Let $f(x,y)=x^2-2xy+y^2$. I know this is not coercive as along the line $y=x$, when $||x|| ->\infty, f(x,x)=0$. But I don't understand what is wrong with the following way. $$f(x,y)=x^2-2xy+y^2=...
2
votes
2answers
156 views

Use of partial derivatives as basis vector

I am trying to understand use of partial derivatives as basis functions from differential geometry In tangent space $\mathbb{R^n}$ at point $p$, the basis vectors $e_1, e_2,...,e_n$ can be written ...
0
votes
1answer
69 views

Getting an idea to prove the equality of mixed partial derivatives $f_{xy}=f_{yx}$

I was reading the partial differentiation unit today and came across this theorem. I read this before but never focused towards the proof. Now while seeing the proof, I am thinking how would one get ...
0
votes
1answer
39 views

Help maximizing this 14 variable scalar valued function?

I have never done multivariable calculus and the last time I did any calculus was 6 years ago in high school. This is just a problem related to something else I'm trying to solve just for fun. I want ...
1
vote
1answer
617 views

how to evaluate this integral in simple form?

$$f (x) =\int_{0}^{x} \int_{-\infty}^{\frac{s^2}{2}} e^{s-\frac{t^2}{2}}\, dtds$$
0
votes
3answers
38 views

Using Divergence Theorem to evaluate the integral

Find the value of $$ \iint_{\Sigma} <x, y^3, -z>. d\vec{S} $$ where $ \Sigma $ is the sphere $ x^2 + y^2 + z^2 = 1 $ oriented outward by using the divergence theorem. So I calculate $ div\vec{F}...
3
votes
1answer
95 views

Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: $$\int\...
2
votes
1answer
29 views

Formal construction of line and surface integrals

I´m currently studying these topics, but I think that there is a lack of formalism in the books that I´m reading: Apostol vol 2; Marsden; Thomas vol 2. So I would really appreciate if you can ...
1
vote
1answer
470 views

Divergence of a radial $1/r^2$ vector field

How to obtain the divergence of the function $F(r,\varphi,\theta)=\hat{r}/r^2$ where $\hat r$ is the unit vector in radial direction? Is there a solution without computing the surface integral for ...
0
votes
1answer
33 views

Why is the directional derivative maximal in the direction of gradient?

As far as I understand, if I am given a function of the form $w=f(x,y,z)$ (that represents the temperature for example) and I want to find what is the direction I should walk (if I am at $(x_0,y_0,z_0)...
2
votes
0answers
48 views

Matrix inverse series expansion

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from $\...
2
votes
2answers
159 views

Notation for i, j, k component of gradient vector at point x, y, z?

I have a function, $w=x^2+y^2-z^2$, and its gradient vector, $\nabla w=(2x, 2y, -2y)$. How can I write the equation for its tangent plane? Is something like the following accurate? $$ p=\nabla w_\hat ...
4
votes
2answers
69 views

How to compute the area of the portion of a paraboloid cut off by a plane?

How to compute: The area of that portion of the paraboloid $x^2+z^2=2ay$ which is cut off by the plane $y=a$ ? I think I have to compute $\iint f(x,z) dx dz$ , where $f(x,z)=\sqrt{(x^2+z^2)/2a}...
1
vote
1answer
43 views

Convert to polar and evaluate

I have $$z= x^2 + y^2$$ $$z=2x$$ I set them equal to get their intersection, I get $$2x= x^2 + y^2$$ $$0= x^2 -2x +y^2$$ by completing square I get $$y= \pm \sqrt{1-(x-1)^2}$$ I need to put ...
2
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0answers
43 views

How to compute the area of that portion of the conical surface $x^2+y^2=z^2$which lies above the $xy$-plane and is cut off by a sphere ? [closed]

How to compute the area of that portion of the conical surface $x^2+y^2=z^2$which lies above the $xy$-plane and is cut off by the sphere $x^2+y^2+z^2=2ax$ ?
2
votes
2answers
58 views

Find the volume between two paraboloids

Find the volume of the solid enclosed by the paraboloids $z = 1-x^2-y^2$ and $z = -1 + (x-1)^2 + y^2$. Using triple integrals, it is known that $V = \iiint_R \mathrm dx\,\mathrm dy\,\mathrm dz$, and ...
2
votes
1answer
26 views

Finding the directional derivatives of this function at the origin

I was having trouble finding the directional derivatives at the origin of the function $$f(x,y)=\begin{cases} [(2x^{2}-y)(y-x^{2})]^{1/4} & \text{for $x^{2} \leq y \leq 2x^{2}$} \\ 0 ...
3
votes
1answer
2k views

What is the meaning of evaluating the divergence at a _point_?

Reading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...
0
votes
1answer
19 views

Langrange multiplier, confusion at setting up the problem

Question Find the rectangular box with the largest volume that fits inside the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, given that it sides are parallel to the axes Solution Clearly the box will have ...
0
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0answers
21 views

Convergence of integrals if curve parametrisations converge

Let $\boldsymbol{r}:[a,b]\to\mathbb{R}^3$ be the piecewise continuously differentiable parametrisation of a piecewise smooth curve. If, for all $n\in\mathbb{N}$, $\boldsymbol{r}_n:[a,b]\to\mathbb{R}^...
4
votes
1answer
286 views

What's the differences between multi variable and vector calculus

This is a conceptual question. If we use vector calculus and multi variable calculus as synonym, will it be completely wrong? If so what topics does multi variable calculus have but vector ...
3
votes
1answer
60 views

Why is divergence defined as $\mathbf{\nabla} \cdot \mathbf{v}$?

Suppose I am working in $\Bbb R^3$. Suppose I have a pond and I drop some dust on the surface. If the materials spread out, I have positive divergence, usually. Let $\mathbf{v}(\mathbf{x})$ denote ...
0
votes
1answer
20 views

What are the limits for this triple integral?

This is probably a very easy/silly question, but still I'm not sure about it. I want to calculate the volume of a body bound between the graph of $x^2+y^2-z^4=1$ (what does it look like?) and the ...
7
votes
2answers
395 views

Derivative of position [duplicate]

[Beginning calculus question.] I saw in a calculus lecture online that for a position vector $\boldsymbol{r}$ $$\left|\frac{d\boldsymbol r}{dt}\right| \neq \frac{d\left| \boldsymbol r \right|}{dt}$$ ...
1
vote
2answers
27 views

Surface integrals and conditions for surfaces

I have researched online for the conditions that are required for a surface to be called smooth. One of which is that the derivative matrix of the map $F$, mapping a region from $\mathbb{R}^2$ to $\...
1
vote
1answer
68 views

Is $\int_0^2 [\int_x^ {\sqrt 3x} f(\sqrt{x^2+y^2})dy]dx=\int_{\pi /4}^{\pi /3}[\int_0^{2\sec \theta}rf(r)dr]d\theta$?

Is it true that $\int_0^2 [\int_x^ {\sqrt 3x} f(\sqrt{x^2+y^2})dy]dx=\int_{\pi /4}^{\pi /3}[\int_0^{2\sec \theta}rf(r)dr]d\theta$ ? . I can't figure out what polar co-ordinate transformation would it ...
0
votes
2answers
65 views

Path of a cycloid

In this question, it's said that the path of a cycloid can be given as this parametric equation: $$\begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*}$$ and is shown here: ...
2
votes
1answer
39 views

A use of Implicit Function Theorem

Honestly, I don't understand the question well. As a try, I defined a function $G:\mathbb{R}^{2+1}\rightarrow \mathbb{R}^2$ where $G(x,y,z)=(f_1(x,y,z),f_2(x,y,z))$ so that $f_1(x,y,z)=x-y$ and $f_2(x,...
3
votes
2answers
54 views

Minimal c satisfying $x+y-(xy)^c \geq 0$ for all $x,y\in [0,1]$

What is the minimal real $c$ satisfying $x+y-(xy)^c \geq 0$ for all $x,y \in [0,1]$? Experimentally (though my experiments weren't necessarily accurate enough) I reached as low as $c=\tfrac{13}{32}$, ...
2
votes
1answer
32 views

Einstein Summation with Del Operator

Can someone show explicitly me why $2B_k\nabla B_k = \nabla B^2$ ? Is $B_k\nabla B_k$ just $B_x\nabla B_x+B_y\nabla B_y+B_z\nabla B_z$? But then I end up with nine terms on the LHS and I can't ...
0
votes
1answer
35 views

How do I apply ∂/∂x in these examples?

If we have a function $ f(x,y(x)) $ it is clear to me that $$ \frac{df}{dx}=\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx} \tag{A} $$ so $ \frac{df}{dx} \neq \frac{\...
4
votes
2answers
3k views

Computing the derivative of a quadratic form and matrix chain rule

I'm working on using the Generalized Method of Moments to analyze some yogurt purchase data, and in the course of trying to implement the standard Hansen method (i.e. not an empirical likelihood ...
3
votes
1answer
38 views

Integration involving Inner Product

Suppose $f: {\bf R}^n \to {\bf R}^n $be a continuous function such that $\int_{{\bf R}^n} \vert f(x) \vert \, dx < \infty$. Let $ A \in GL_n({\bf R})$. Show that $$ \int_{{\bf R}^n} f(Ax) e^{i\...
1
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0answers
52 views

For parameter find the number of solutions

We have $\begin{cases} x^3-3xy^2-tx=-2\\3x^2y-y^3-ty=2\end{cases}$ for every $t \in \mathbb{R}$ determine the number of points $(x,y)=(x(t),y(t))\in \mathbb{R^2}$ which satisfy our system of ...
1
vote
1answer
35 views

Volume of a Sphere Segment Cut Out by Zigzag Motion of Radius Vector Through Sphere Center

A closed loop curve on a sphere radius $R$ enclosing curved area $A$ subtends a solid angle $ A/R^2$ steridians at center of the sphere. Volume $V$ is enclosed by concurrent straight generators ...
1
vote
1answer
59 views

What's the $\otimes$-operator in the proof of Reynolds' transport theorem at Wikipedia?

In the proof of Reynolds' transport theorem at Wikipedia, they use the identities $$\nabla\cdot(v\otimes w)=v(\nabla\cdot w)+\nabla v\cdot w$$ and $$(a\otimes b)\cdot n=(b\cdot n)a\;,$$ where $n$ is ...
0
votes
1answer
57 views

Solving a PDE with constant initial and boundary conditions

Consider the PDE: $$ u_t + (1-2u) u_x =0 \text{, where } x<0 \text{ and } t>0,$$ with initial and boundary data given by $$u(x,0) = \frac{1}{4} \text{ for } x<0, \text{ and } u(0,t) = 1, \...
3
votes
1answer
34 views

Solving by limit definition

To show that $\displaystyle\lim_{(x,y)->(0,0)} \dfrac{5x^2y}{x^2+y^2}=0$ I tried to do this by the limit definition by pluging 0 to l in lim(x,y)->(Xo,Yo) | f(x,y)-l |. Then i got stucked where i ...
0
votes
0answers
7 views

What is the generic way to find a normal from curvilinear coordinates for any amount of dimensions?

For a surface $\vec{x} \in \partial C$ that wraps around a volume $\vec{s} \in C$ parameterised by coordinates $\vec{u}$ the surface normal $\hat{n}$ is $\frac{\partial\vec{x}}{\partial u_0} \times \...
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0answers
27 views

How do I notate the gradient with respect to an arbitrary parameter?

The gradient is: $ \nabla y = \sum_{i=0}^N{\frac{\partial y}{\partial s_i}} \hat{e_i} $ where $\vec{s}$ is some space parameter. I'd like to find and write down the value $\sum_{i=0}^N{\frac{\...
0
votes
2answers
542 views

Double Integrals

$(a)$ Sketch the region of integration in the integral $$\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} dx dy$$ By changing the order of integration, or otherwise, evaluate the ...
1
vote
0answers
32 views

Surface Integral Over a Triangle with Given Vertices

My question is about how to define the bounds for the integral. given a vector field $F$ and a triangle with vertices $(1,2,3), (4,5,6),(7,8,9)$. how to define the bounds when integrating the vector ...
5
votes
3answers
34 views

Show that $\lim\limits_{r\to\infty} \textrm{\{} \|x-ru\|-\|y-ru\| \textrm{}\} = \left<y-x,u\right>$

Let $x,y,u \in \mathbb{R}^2, r\in\mathbb{R}$ and $\|\cdot\|$ be the norm. Show that $$\lim\limits_{r\to\infty} \textrm{\{} \|x-ru\|-\|y-ru\| \textrm{}\} = \left<y-x,u\right>$$ I have to tried ...
1
vote
1answer
87 views

Why is divergence related to “volume”? [duplicate]

$$\mathrm{div}\ {\bf F} = \lim_{V\to\{0\}}{1\over|V|}\oint_V {{\bf F}\cdot d{\bf a}}$$ Wikipedia says: In physical terms, the divergence of a three dimensional vector field is the extent to ...
2
votes
3answers
63 views

To find the volume of a certain solid cone

A solid cone is obtained by connecting (with a line segment in $3$ dimensional Euclidean space ) every point of a plane region $S$ with a vertex not in the plane $S$ . Let $A$ denote the area of $S$ ...
2
votes
2answers
31 views

Definite or indefinite integral

[Beginning calculus question.] From Edwards and Penney (6e) p. 810, worked Example 8: Suppose that a moving point has given initial position vector $\boldsymbol{r}(0) = 2\boldsymbol{i}$, initial ...
-1
votes
1answer
46 views

Let $S:=\left\{(x,y):|x|+\left|y\right|\le1\right\}$; then how to evaluate $\displaystyle{\int\int_S e^{x+y}\,\operatorname{d}x\,\operatorname{d}y}$?

Let $S:=\left\{\left(x,y\right):\left|x\right|+\left|y\right|\leq 1\right\}$. How to evaluate $$ \displaystyle{\iint_S e^{x+y} \, \operatorname{d}x \,\operatorname{d}y}? $$ Please help. Thanks in ...
1
vote
1answer
25 views

Show the Jacobian back to itself is 1

I'm working through problems for revision but I've lost the solutions. Show that $\frac{\partial (x, y)}{\partial (u, v)} \frac{\partial (u, v)}{\partial (x, y)} = 1$. My attempt at a solution: ...
1
vote
1answer
26 views

Maximum increase of a function along a surface

I'm not quite sure how to approach this question - it's a practice problem, and there is no answer key. Someone told me the answer was to project the gradient onto the surface, but I'm not sure how to ...