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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0answers
29 views

Bound of integration over the surface area?

Compute the surface area of that portion of the sphere $x^2+y^2+z^2=a^2$ lying within the cylinder $x^2+y^2=ay$ where $a>0$ I first parametize the sphere using spherical coordinate. I think ...
1
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1answer
31 views

Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor

I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y). Eventually I managed to pull the (presumably) correct ...
-3
votes
1answer
50 views

Find the directional derivative using $f(x,y,z)=xy+z^2$. [closed]

Find the directional derivative using $f(x,y,z)=xy+z^2$, at the point $(2,3,4)$ in the direction of a vector making an angle of $\frac{3\pi}{4}$ with grad $f(2,3,4)$. PS - I am having trouble ...
1
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0answers
42 views

Proof that any unit-speed-reparametrization of a curve preserves orientation and is an inverse of an arc length function based at some $t_0$.

I am not able to prove the following two facts about a unit speed reparametrization of a curve. Let $\alpha$ be defined on some interval $I$ and define for $t_0\in I$ ...
0
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2answers
40 views

Find the multivariable limit with exponential

Let $f(x,y): \mathbb{R^2} \to \mathbb{R}$ where $f(x,y) = \frac{e^{xy^2}-1}{x^2+y^2}$. If it exists, find $$\lim_{(x,y) \to (0,0)}\frac{e^{xy^2}-1}{x^2+y^2} $$I'm not entirely sure how to approach ...
1
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2answers
64 views

Determine whether the integral converges or diverges

Determine whether the following integral converges or diverges: \begin{align*} \iint_Q e^{-xy} \ dA, \end{align*} where $Q$ is the first quadrant of the $xy$-plane. How should I go about this ...
0
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0answers
41 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
0
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1answer
19 views

probability of joint PDF

I found $k = 4$ and yes, the are independent. But for the last one I know how to find the probability if they are like $x$ from $0$ to a number and $y$ from $0$ to a number so the limit of double ...
1
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0answers
25 views

How do I find $C^1$ mapping with given range

I have a following question. Find a one-to-one $C^1$ mapping $f$ from the first quadrant of the $xy$-plane to the first quadrant of the $uv$-plane such that the region where $x^2 \leq y \leq 2x^2$ ...
-1
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1answer
35 views

Prove $f$ differentiable: $\|f(v)\| \leq M \cdot \|v\|^{N+1}$

Let $f:\Bbb R^n\to \Bbb R^m$. Assume that there exists $M\in\Bbb R$ positive such that for all $v \in \mathbb R^n$: $$ \|f(v)\|\leq M\cdot \|v\|^{N+1} $$ for some natural number $N\geq 1$. Prove $f$ ...
-1
votes
1answer
21 views

Is $Cr^a\hat r$ always a conservative vector field?

Is the vector field $\vec r(r, \theta, \phi) = Cr^a\hat r$, where $C, a\in \Bbb R$ are constants and $r \in \Bbb R^+ \cup \{0\}$ is the radial component, always a conservative vector field? I really ...
6
votes
1answer
79 views

Find multivariable limit $\frac{x^2y}{x^2+y^3}$

Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a ...
3
votes
1answer
35 views

Lipschitz condition not satisfied

To show there is no contradiction to existence and uniqueness $\displaystyle\frac{|f(x,u)-f(x,v)|}{|u-v|}= \displaystyle\frac{|x||u^{1/2}-v^{1/2}|}{|u-v|}=\frac{|x|}{u^{1/2}+v^{1/2}}$ I understand ...
1
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2answers
440 views

Unit speed reparametrization of curve

I am learning Elementary Differential Geometry by O'Neill and having a hard time with this exercise. Suppose that $\beta_1$ and $\beta_2$ are unit-speed reparametrizations of the same curve $\alpha$. ...
3
votes
4answers
100 views

Finding a limit with two independent variables

I must find the following limit: $$\lim_{(x,y)\to (0,0)}\frac{x^2y^2}{x^2+y^2}$$ Substituting $y=mx$ and $y=x^2$, I have found the limit to be $0$ both times, as $x \to 0$. I have thus assumed that ...
0
votes
1answer
66 views

Calculate the integral of $f(x,y,z)=e^z$ over the tetrahedron with vertices $(4,0,0), (0,4,0), (0,0,6)$

How does one setup this integral? Do I simply need an equation for the plane? I want to say this is the answer $\int_{0}^{6}\int_{0}^{4-z}\int_{0}^{4-y-z}(e^z)\space{dx}\space{dy}\space{dz}$ but ...
1
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1answer
42 views

Let S be a parallelogram not parallel to any of the coordinate planes. Area of S?

Let S be a parallelogram not parallel to any of the coordinate planes. Let $S_1,S_2,S_3$ denote the areas of the projections of S on the three coordinates planes. Find the area of S in terms of ...
1
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1answer
28 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
0
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0answers
30 views

Fundamental Theorem of Calculus for Line Integrals

Use the Fundamental Theorem of Calculus for Line Integrals to compute $\int_C F*dr$ where $$F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy-2z)k$$ and C is the path from $(1,6,-1)$ to $(5,2,3)$ given by $x(t)=2t+1, ...
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1answer
41 views

Vector operator

With the help of Vector Operator $\nabla$ and the rules of differentiation and multiplication of vectors, prove the following identities: $$grad (\varphi \psi) = \varphi grad(\psi) + \psi ...
1
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1answer
45 views

Gradient of Vector

What would be the gradient of following vectors in Cartesian Co-ordinate? $$grad (\vec{c} \cdot \vec{r})$$ where $\vec{c}$ is the constant vector and $\vec{r}$ is the radius vector. Thanks.
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0answers
31 views

Computing the integral

Let $A=(a_{ij})_{1 \leq i,j \leq n}$ be a positive definite and symmetric matrix of dimension $n \times n$. I want to compute $$\int_{\mathbb{R}^n} \exp( -\frac{1}{2} \sum_{1 \leq i,j \leq ...
1
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1answer
53 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
0
votes
1answer
49 views

How to visualize a line integral

I was studying for multivariable calculus and I came across the line integral section. Visually, I perfectly understand why $\int_{t_1}^{t_2} f(x(t),y(t))s'(t)\,dt$ computes the area under $f(x,y)$ ...
2
votes
1answer
111 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
1
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1answer
111 views

Show that f is not differentiable at the origin of the following function.

Show that f is not differentiable at the origin of the following function: $f(x,y) = \left\{\begin{matrix}\frac{2xy}{x^2+y^2}, (x,y) \neq (0,0)\\ 0, (x,y) = (0,0) \end{matrix}\right.$ I was thinking ...
1
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4answers
84 views

evaluating the limit $ \lim_{p \rightarrow (0,0) } \frac{x^2y^2}{x^2 + y^2} $

I was trying to evaluate $$ \lim_{p \rightarrow (0,0) } \frac{x^2y^2}{x^2 + y^2} $$ but only partially managed to do so through implying $y=kx$ but didn't find any other appropriate functions for ...
1
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3answers
56 views

Does the limit exist? (Calculus)

Consider the function $$f(x,y)=\frac{2xy^2\sin^2(y)}{(x^2+y^2)^2}.$$ Does the limit exist when $(x,y)$ tends to $(0,0)$?
4
votes
4answers
60 views

Is this a valid proof? Find $\lim_{(x,y) \to (0,0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$

I "solved" this limit using polar coordinates, but my question is - is this a definite proof that the limit exists? Or maybe there is some path that I missed when I transformed to polar coordinates? ...
1
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0answers
32 views

The free surface of the wave is a material surface

If we define the free surface by: $F(x,y,t)=y-h(x,t)=0$ Then for this to be a material surface $\frac{DF}{Dt}=0$ on $y=h(x,t)$ However on $y=h(x,t)$, $F=0$, so doesn't this just imply ...
35
votes
7answers
46k views

What is Jacobian Matrix?

What is Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone explain with examples? Thanks! :)
1
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1answer
44 views

Derivative with respect to the normal?

I am trying to use greens theorem to show the following: $$\int \int (f_{xx}+f_{yy}) \, dx \, dy=\int\frac{\partial f}{\partial n} \, ds$$ I am not completely sure how to treat the $d/dn$. I have ...
0
votes
1answer
37 views

Inverse Jacobian Matrix

I have a question about using the Jacobian matrix with the Newton-Raphson method. The regular Newton-Raphson method is initialized with a starting point $x_0$ and then iterated ...
0
votes
1answer
67 views

Trapezoidal rule in 2 dimensions

I'm using trying to integrate a function in MATLAB using the trapezoidal rule. I'm struggling to get the limits right and how to set up the steps. The limits for $x$ are $[0,2]$ and the limits for ...
0
votes
0answers
15 views

If $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$

Let $f:\mathbb R^2\to \mathbb R$ if $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$ My attempt: Let $\epsilon>0$, we know that $\exists M>0$ such that ...
3
votes
3answers
427 views

derivative of exponential of matrix trace

What is the derivative of $\sum_{ij}e^{-d_{ij}^2(X)}=\sum_{ij}e^{-\operatorname{tr}(X^TC_{ij}X)}$, w.r.t $X$ where $C_{ij}$ is a constant matrix and $d_{ij}^2(X)$ denotes the squared Euclidean ...
1
vote
1answer
58 views

Limit and continuity

Hello I am a bit confused about this problem. It says, define a function f over the whole plane as $$f(x,y)=0$$ if $x=0$ and $$f(x,y)=0$$ if $y=0$ other wise defined by ...
0
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0answers
28 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
1
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1answer
54 views

A $2$-variable non-differentiable function whose partial derivatives exist

If a $2$-variable function is not differentiable at some single point, but has finite partial derivatives for both variables at that point, can it also have a derivative in any direction, I mean is it ...
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1answer
35 views

derivative of a linear mapping

What is the derivative of a linear mapping A: R^n -> R^n? I assume it must be a tensor. In particular, if I have a linear function of a vector x, A(x), what is DA(x)?
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0answers
18 views

Contour lines and implicit function theorem

$f = y^2-x^2+x^3$ I want to find all points $(x_0,y_0)$ such for some neighbor set $f(x,y) = f(x_0,y_0)$ is set $x=\varphi(y), \varphi \in C^{\infty}, \varphi(y_0)=x_0$.
3
votes
4answers
103 views

multivariable limit of $\frac{x^2-y^2}{\sqrt{x^2+y^2}}$

Calculate multivariable limit of $$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2-y^2}{\sqrt{x^2+y^2}}$$ How to do that? I was trying to figure out any transformations e.g. multiplying by denominator but I ...
1
vote
1answer
37 views

Limits and 'L'Hopital' in higher dimensions.

Let $f,g:\mathbb{R^n}\rightarrow\mathbb{R}$ be two $C^2$ functions with a critical point at $x_0$ and $f(x_0)=g(x_0)=0$ and $D(Df(x_0))=cD(Dg(x_0))\not= 0$, where $c$ is a constant. Show that ...
2
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1answer
171 views

Why is the composition of smooth multivariable functions smooth?

Is an an easy way to see that if $f: U \subset \mathbb R^n \rightarrow V \subset \mathbb R^m$ and $g: V \subset \mathbb R^m \rightarrow \mathbb R^p$ are smooth functions then their composite $g ...
1
vote
1answer
55 views

Drawing 3D level sets/curves

I was asked to Draw the level sets of $$f(x,y,z) = 9x^2-4y^2-36z$$ I under how to draw them with just two variables, but does anyone have any resources that could help me visualize drawing these ...
2
votes
1answer
28 views

Inequality with Laplacian

I'm trying to solve a larger problem and have reduced it to showing that $$\int_\Omega (u-v) \Delta v dx \geq 0$$ Here $u, v$ are both continuous on $\overline\Omega$, $C^2$ on $\Omega$, which is an ...
0
votes
1answer
19 views

Having trouble setting up the limits of integration.

Let $E$ be the solid below the plane $z=8$ and above the cone $z=\sqrt{x^2+y^2}$. Find the mass of $E$ if the density $\rho(x,y,z)=z$. I'm supposed to use triple integrals with cylindrical ...
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votes
1answer
110 views

$\epsilon-\delta$ proof of a limit of a function $z(x,y)$

I was working on this exercise to prove the differentiability of a function at a certain point, but I got stuck in proving the following limit. $\lim_{(x,y)\rightarrow (0,0)} ...
0
votes
1answer
65 views

Flux integral through elliptic cylinder

Find the flux of F=(3x,2y,z) through the volume bound by the xy plane, the elliptic cylinder (x/3)^2+(y/2)^2=1, and the paraboloid x^2 + y^2 =z, and hence find the components of the flux through the 3 ...
-1
votes
1answer
48 views

multivariable limit of $\frac{x^2y^2}{x^2y^2+(x-y)^2}$ [closed]

Show that limit of $f(x,y)=\lim_{(x,y) \rightarrow (0,0) } \frac{x^2y^2}{x^2y^2+(x-y)^2}$ does not exist but show as well the following holds: $\lim_{x \rightarrow 0}\left( \lim_{y \rightarrow 0} ...