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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2answers
329 views

Converting a point to Cylindrical and Spherical Coordinates

How is any point on the Cartesian coordinates converted to cylindrical and spherical coordinates. Taking as an example, how would you convert the point (1,1,1)? Thanks in advance.
1
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1answer
67 views

How to find all local extrema using multi-variate calculus

Find all local extrema of $f(x, y) = x^2 + y^2$. That is, find their locations and values. I started this problem but when I took the first derivative I got the critical points as $(0, 0)$. Then ...
1
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1answer
96 views

Interchange of partial derivative and limit

Consider the following expression: $$\frac{\partial}{\partial m} \lim_{T \rightarrow \infty} \gamma(T,m)$$ where $\gamma$ is a function of $T$ and $m$. My question is just: can I permute the partial ...
1
vote
1answer
137 views

Surface integrals of second kind

In the formula for calculating surface integrals of second kind, we have: But, this integral is denoted by $\int \int _S \vec{F}\cdot \hat{n}dS $ . So, should we always normalize the expression $ ...
2
votes
1answer
75 views

For given $p$, a regular mapping $F:M \to N$ of surfaces can be diffeomorphism

Want to show : For given point $p$ of M, a regular mapping $F:M \to N$ of surfaces has a neighborhood $U$ such that $F|_{U}$ is a diffeomorphism of $U$ onto a neighborhood of $F(p)$ in N. I learned ...
3
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0answers
76 views

Tangent bundle of a surface is a manifold

My differential geometry textbook defined the tangent bundle of a surface as the set of all tangent vectors to M at all points of M. The abstract patches are also given : ...
1
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1answer
97 views

A question concerning partial derivatives

Let $F(x-y,y-z,z-x)=0$,find $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$. This is a homework problem,I don't know how to do,appreciate any help.
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2answers
56 views

Area of a parallelogram using Cross Product

How do you compute the area of the parallelogram with 4 arbitrary corners, say at (1,1,1), (2,3,4), (3,2,1), and (4,4,4) using a cross product? I understand with 3 corners but getting a little lost ...
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5answers
60 views

what is this value called?

I've come across this operation that takes in two vectors $a=(X_{a},Y_{a})$ and $b=(X_{b},Y_{b})$ and returns a single number $X_{a}\cdot Y_{b}-Y_{a}\cdot X_{b}$ Is this like 'a thing'? Like the ...
0
votes
1answer
36 views

Check for directional derivative and show that $f(1,1,1) > f(0,0,0)$ when given the partial derivatives

I'm kinda lost in this exercise Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be of class $C^1$ and $\forall x=(x_1,x_2,x_3)\in\mathbb R^{3}$ $$\frac{\partial f}{\partial x_{1}}(x)=x_{2}, ...
1
vote
1answer
177 views

Check for differentiability, continuitity and existence of partial derivatives/directional derivative

The exercise is : Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be as follows : $$f(x,y) = u(x,y)\cdot(x^2+y^2)$$ where $$u(x,y) = \begin{cases} 1 &\text{for } x=0 \vee y=0 \\0 &\text{for ...
1
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2answers
112 views

How to find this mass?

Let $R$ be the region in the first quadrant of the plane bounded by the lemniscates of the following equations: $\rho^2=4\cos(2\theta)$, $\rho^2=9\cos(2\theta)$, $\rho^2=4\sin(2\theta)$, and ...
2
votes
1answer
436 views

Using the arc length function to find a parameterization of C in terms of s

The problem: Find the arc length function $s(t)$ for the curve defined by $\vec r(t)$. Then use this result to find a parametrization of $C$ in terms of $s$. $$\vec r(t) = a\cos^3t\,\hat i + ...
2
votes
2answers
9k views

Normal Vector to a Sphere

I'm having kind of a problem on calculating the normal vector to a sphere using a parameterization. Consider a unit-radius sphere centered at the origin. One can parameterize it using the following: ...
0
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0answers
17 views

Definition of a multiple integral without hyperrectangles?

Could you define multiple integrals in a formally satisfactory manner without resorting to hyperrectangles and limits of integral sums? I just don't like that definition for no serious objective ...
4
votes
2answers
53 views

Find $u'(0)$ where $u(t) = f(2009t, t^{2009})$ given the differential of $f$ in $0$

I'm having trouble with this exercise, probably don't have enough theoretical info. How do I approach this? Let $f : \mathbb{R}^2\rightarrow \mathbb{R}$ where $f$ - differentiable in $0$ and ...
4
votes
3answers
200 views

Find maximum and minimum of $f(x, y) = xy$ on $D = \left\{ (x,y) \in \mathbb{R}^2: x^2+2y^2 \leq 1 \right\}$

I'm kinda stuck on this one : Find the minimum and maximum of the given function $f$ on $D$, where $$f(x, y) = xy$$ and $$D = \left\{(x,y) \in \mathbb{R}^2 : x^2+2y^2 \leq 1 \right\}$$ I don't ...
4
votes
1answer
243 views

In a Frenet-Serret frame, what are $\Delta\vec T$ and $(\vec a\vec\nabla)\vec T$

Given a Frenet-Serret frame $(\vec T(t), \vec N(t), \vec B(t))$ defined by a curve $\vec \gamma(t)$ with $$\begin{array}{rcl} |\tfrac{d}{dt}\vec\gamma(t)| &\equiv& 1, \\ \vec T(t) ...
2
votes
1answer
32 views

For 1 form $\xi$, $F^{*}(d\xi)=d(F^{*}\xi)$

Let $F:M \to N$ be a mapping of surfaces, and $\xi$ be a function. I want to show the following identity. $$F^{*}(d\xi)=d(F^{*}\xi)$$ What I did : Fix a tangent vector $v$. ...
4
votes
1answer
88 views

Which integral theorem to use to evaluate this triple integral?

Take the normal pointing outwards from the surface. Use an appropriate integral theorem $$\iint_S \textbf{F}\cdot d\textbf{S} \space \space where \space \space \textbf{F} (x,y,z)=(x^3,3yz^2,3y^2z+10) ...
1
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1answer
77 views

Geometrical explanation of the Cross Product

Teaching myself multivariable calculus before class starts in August and got stuck with the following two questions which I have no one but you to ask. So any deep explanation would be great! Show ...
1
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1answer
95 views

Determining $u=v \times w$ using the cross product

Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula. What are the possible angles $x$ between two unit vectors ...
2
votes
1answer
108 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
3
votes
1answer
444 views

How to show that the Hessian matrix of $G$ is positive definite?

Let $\{g_i:X\subset\mathbb{R}\rightarrow\mathbb{R};\;i=1,...,m\}$ be a linerly independet set of real functions. Given $n$ points $(x_1,y_1),...,(x_n,y_n)\in X$, consider the following function ...
4
votes
2answers
203 views

minimum of this function

Define $f(x,y) = (a - bx - by)e^{-(x^2+c y^2)}$, where $c > 1$ and $b>a$ then define $g(x,y) = f(x,y)$ if $x<y$ and $g(x,y)=f(y,x)$ if $x>=y$. Look at the function $g(x,y)$. ...
0
votes
1answer
166 views

Jacobian determinant and orientation

So in Jacobian determinant, it is often said that it gives information about whether Jacobian matrix changes orientation, but I cannot get what orientation exactly in this context.
1
vote
1answer
598 views

Electric field of finite sheet: Full analytical solution of integration?

I am trying to work out the integral $$E_{z}(x,y,z)=\alpha\int\int\frac{z\, dx'\, dy'}{((x-x')^{2}+(y-y')^{2}+z{}^{2})^{3/2}}$$ with the limits $$-\frac{a}{2}\leq ...
0
votes
2answers
98 views

Local extrema of a function subject to an inequality

Preparing for my exams I came across a problem I don't know how to solve : Find the extremums of given function on domain $D$ and check if the function reaches it : $$f(x, y, z) = x + y + z$$ ...
5
votes
1answer
245 views

Definition of smoothness “up to boundary”

Let $U\subseteq \mathbb{R}^n$ be an open set and let $f\in\mathcal{C}^k(U)$ for some positive integer $k$. Are the following definitions of $\mathcal{C}^k$ regularity "up to boundary" equivalent? ...
3
votes
3answers
115 views

Differentiation chain rule

If $f(tx,ty,tz) = t^nf(x,y,z)$ then in my lecture notes it says differentiating this equation with respect to $t$ gives: $$x\frac{df}{dx} + y\frac{df}{dy} + z\frac{df}{dz} = nt^{n-1}f(x,y,z)$$ But ...
2
votes
2answers
385 views

How to find a solution(s) when given two equations of two variables?

This is something that I have been trying to comprehend from an algebraic point of view. Take the equations $$3x^2 - 12y = 0$$ $$24y^2-12x = 0 $$ The first equation implies that $x =\pm 2\sqrt{y}$ and ...
0
votes
1answer
142 views

Surface integral of a scalar

Evaluate the following integral $$\int \int_S z^2 dS$$ where $S$ is the surface of the cube $[-1,1] \times [-1,1] \times [-1,1]$ My thoughts I'm quite lost here. How do I know the projection ...
1
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2answers
95 views

Integral of a vector field dotted with a unit normal

Find $\int_C{F \cdot \hat n ds}$ where $F= (2xy,-y^2)$ and $\hat n$ is the unit outward normal to the curve C in the xy-plane and C is the ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ traversed in the ...
3
votes
2answers
976 views

Gauss's divergence theorem for a scalar field

I know Gauss's divergence theorem for a vector field: $$\iint{\vec{F}\cdot\hat{n}}\space{dS}=\iiint\nabla\cdot\vec{F}\space{dV}$$ But how do you apply this to a scalar field? For example, if you ...
2
votes
1answer
60 views

Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$

I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to $$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$ What now?
0
votes
2answers
128 views

Bijection between $\mathbb R$ and $\mathbb R^N$

Hi, could you give me an example of a bijection between $\mathbb R^T$ and $\mathbb R$ with $T$ being a positive integer which is as regular as possible like $C^{\infty}$ or Lipschitz continuous at ...
4
votes
1answer
150 views

Integrating by using change of variables and by making a substitution

Let $D$ be the region bounded by $x=0$, $y=0$, $x+y=1$ and $x+y=4$. Evaluate $$\iint_D \frac{dx\,dy}{x+y}$$ by making the change of variables $x=u-uv$, $y=uv$ My attempt I understand I must ...
0
votes
2answers
84 views

Partial differentiation chain rule

I don't understand why d(phi)/d(x_i) is not d(phi)/d(x_i + st_i) since if I have f(s) = g(x,y,z), then the partial derivative of f would be df/ds = dg/dx*dx/ds + dg/dy*dy/ds + dg/dz*dz/ds using the ...
4
votes
1answer
65 views

change of variables using a substitution

Let $D$ be the triangle with vertices $(0,0),(1,0)$ and $(0,1)$. Evaluate $$\iint_D \exp\left( \frac{y-x}{y+x} \right) \,dx\,dy$$ by making the substitution $u=y-x$ and $v=y+x$ My attempt ...
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vote
2answers
1k views

What does surface integral define?

What does a surface integral define.... what is the difference between $dS$ and $dA$ if any..? I know that $dA$ give the volume of an function with 2 variables, but what about $dS$ does it do the same ...
1
vote
1answer
45 views

Is there an easier way to integrate this hectic integral?

Using triple integrals and cartesian coordinates, find the volume of the solid bounded by $$z=3x^2+3y^2-7 $$ and $$z=9-x^2-y^2$$ My take I have equated the two solids and found the ...
6
votes
2answers
398 views

Find the volume using triple integrals

Using triple integrals and Cartesian coordinates, find the volume of the solid bounded by $$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$ and the coordinate planes $x=0, y=0,z=0$ My take I have ...
4
votes
1answer
206 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
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3answers
153 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
4
votes
3answers
835 views

Where is Greens theorem used?

Where is Greens theorem used? I think it's weird going from a vector field to calculating a volume on a scalar field, where do we use this kind of calculation?
0
votes
1answer
49 views

Finding the volume: $(\frac{x}{a}+\frac{y}{b})^2+(\frac{z}{c})^2=\frac{x}{p}+\frac{y}{q}$ ; $x,y,z>0$

Consider the region with $x,y,z>0$ bounded by $$\left(\frac{x}{a}+\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2=\frac{x}{p}+\frac{y}{q}$$ I've tried compute this with a triple integral and ...
2
votes
2answers
104 views

Second point of intersection - parameter removes second solution?

$$\mathbf r_1 (t) = (t^2 - t, t^2 + t) \\ \mathbf r_2(u) = (u+u^2, u-u^2)$$ I'm trying to find two of the intersection points, but I'm lost as to how to approach the question. Is it possible to ...
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votes
1answer
120 views

Continuity in linear maps

Consider the map $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $f(x,y)=(7x+x^4,\ 3x+4y+y^4)$. Then which of the following are true: a) $f$ is discontinuous at $(0,0)$. b) $f$ is continuous at ...
2
votes
1answer
187 views

Surface integral over graph of a function

Q: Evaluate the surface integral $\int$$\int_S$ F.dS, where F(x,y,z)= $3x^2$*i*$-2xy$*j*$+8$*k*, and S is the graph of the function z=f(x,y)=2x-y for $0\leq x\leq2$ and $0\leq y \leq2$. Attempt: ...
2
votes
1answer
97 views

smooth approximate parameterization to polygonal boundary

I can "almost" parameterize the boundary of a square using $${\bf r}(t) = (\cos t)^{1/p} {\bf i} + (\sin t)^{1/p} {\bf j},$$ $0\leq t\leq 2 \pi$, and $p$ is odd. This parameterization is smooth (or at ...