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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0
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2answers
91 views

Proving Multivairble Limit Exists [duplicate]

How do you deal with multivariable limits? We'll use the example $f: \mathbb R ^2 \rightarrow \mathbb R$ $$\lim _{(x,y) \rightarrow (0,0)}\frac{\sqrt{|xy|}}{\sqrt{x^2 + y^2}}$$ The limit doesn't ...
0
votes
1answer
21 views

Limit equivalence

"Let $f:A\subset\Bbb{R}^n\to\Bbb{R}$ be a function and denote $\Bbb{x}=(x_1,\dots,x_n)$ and $\Bbb{p}=(p_1,\dots,p_n)$. Show the following equivalence: ...
0
votes
1answer
49 views

Is this vector identity accurate?

Does this identity hold true for vectors $A$, $B$ and the gradient operator? $(\nabla \cdot A)B = (A\cdot \nabla)B + (B\cdot \nabla)A$
6
votes
2answers
7k views

Gradient of a dot product

The wikipedia formula for the gradient of a dot product is given as $$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$ However, I also ...
0
votes
2answers
20 views

(Inequality) $p \cdot (z-x) \leq \frac{a}{R} | z- x| \Leftrightarrow |p|\leq \frac{a}{R}$

I need to solve this inequality: Let $z \in \mathbb{R}^N$ and $a,R > 0$, prove that $$(\forall x\in B_R(z)) \quad p \cdot (z-x) \leq \frac{a}{R} | z- x| \ \Longleftrightarrow \ |p|\leq ...
0
votes
2answers
64 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
2
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1answer
29 views

Doubt on understanding continuity .

Just preparing for my multivariable-calculus exam and wanted to clear these things: I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the ...
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0answers
16 views

Unit tangent vector

Let $f:I\to \mathbb R^3$ a vector valued function. When we define the unit tangent vector: $T(t)=f´(t)/||f´(t)||$ , $||f´(t)||\neq 0$ is it neccesary that $f$ is a $C^1$ function? or just ...
0
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0answers
22 views

Regarding functions from R² to R: continuity and differentiability

Let $f : U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^2$ is an open set and $P \in U$. I am almost sure the following statements are correct, but please confirm: The only requirement for ...
0
votes
1answer
24 views

Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$

Can anyone help with this: Let $x\in \mathbb R^n$ and $u=f(r)$,where $r=\|x\|$ and f is differentiable . Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$ . I can't ...
3
votes
2answers
818 views

Finding the limits of a multivariable function

Given the following function, determine whether the following function is continuous at $(0,0)$ $$f(x,y)=\begin{cases}\frac{x^2y^2}{x^4+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ ...
0
votes
1answer
61 views

Why i got negative value for volume?

I want to find the indicated volumes under the surface $z=\frac{1}{y+2}$ and over the area bonded by $y=x$ and $y^2+x=2$. After sketching the graph for $x=2-y^2$, and $x=y$ i found that $y=0$ and ...
1
vote
1answer
39 views

Find the point on a parameterized line closest to another line

Let $x_1 = (1, 2, 3)$ and $x_2 = (3, 2, 1)$. Consider the two lines $x_1(s) = x_1 + su_1$ and $x_2(t) = x_2 + tu_2$. $u_1 = (\frac{2}{\sqrt{5}}, \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}})$, $u_2 = ...
1
vote
1answer
29 views

Parameterizing $y = 2 -\sin \frac{\pi x}{2}$

I am trying to parametrize the part of the curve $$ y = 2 -\sin \frac{\pi x}{2} $$ from (0, 2) to (1, 1). I tried the difficult paramaterization $x=t$ and obtained $$ y=2-\sin \frac{\pi t}{2} $$ ...
0
votes
1answer
44 views

Evaluate $ \lim_{ x \to 0, y \to 0}\frac{x^2+y^2+x+y}{x+y}$

How do you find $$\lim \limits_{x \to 0 , y\to 0}\frac{x^2+y^2+x+y}{x+y}$$or prove that it doesn't exist? I've tried every method I know, but I can't find anything conclusive.
0
votes
1answer
31 views

Finding the curl of a cross product

Let $\mathbf{x}$ be the position vector, $\mathbf{a}$ be a constant vector. I need to show that: $$\text{curl}(\mathbf{a}\times\mathbf{x})=2\,\mathbf{a}$$ The problem is, I keep getting ...
1
vote
0answers
34 views

multivarible calculus-directional derivaties

Let $f$ and $g$ be functions from $\mathbb{R}^n \to \mathbb{R}^m$. Assume that $f$ is differentiable at $c$, that $f(c)=0$, and that $g$ is continuous at $c$. Let $h(x)=g(x)f(x)$. Prove that $h$ ...
1
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0answers
31 views

Notation for gradients analogous to partial derivatives

Is there an equivalent of partial differentiation for functions taking multiple vectors as input? I mean the following. If we have a function $f(x,y)$, then a partial derivative is denoted as ...
0
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1answer
33 views

Prove Tetrahedron Opposite Vectors add to $0$

I really need help on this problem, I'm in Multivariable Calculus (Calc III) and I just can't solve this. Let $v_1$, $v_2$, $v_3$, and $v_4$ be vectors whose lengths are equal to the areas of the ...
1
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0answers
44 views

Multivariate calculus (Lagrange multiplier)

If we need to use the method of Lagrange multipliers to find extreme values of a function $f(x, y)$ on a triangle-shaped region in $R ^2$ , how many times would we have to run the method? How many ...
1
vote
1answer
39 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
0
votes
3answers
80 views

construct a path between (-1,0) and (0,2)

So we are given a region S which is above the x-axis and between the semicircle of radius 1 and 2 centred at the origin. we are asked to construct a path that connect the point (-1,0) and (0,2)..and ...
1
vote
1answer
59 views

Computing the limit of a 2-variable function

Show that $\lim \limits _{(x,y)\rightarrow (0,0)} \frac{x^3y}{x^2+y^4}=0$ just using $\epsilon-\delta$ creterion. In fact, choose $\epsilon >0$ arbitary, then we have to find $\delta >0$ such ...
1
vote
1answer
44 views

Verify the divergence theorem for a sphere

Question i cannot work out. I assume you need to get both sides in terms of u and v (parameterized), but im getting pretty confused after completing the first few steps.
0
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0answers
41 views

Directional derivatives in two directions

How can I take a directional derivative in two directions? I mean,$$D_{xy}f(0,0)$$ Because when I have something like $$D_{x}f(0,0),$$ I just use that my direction is in the x axis, $ \vec ...
1
vote
2answers
46 views

How do I find this partial derivative

I have the following function u(x,y) defined as: $$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero, and $u(0,0)=0$ I want to compute its partial derivative $u_{xy}$ at ...
0
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2answers
60 views

The movement of a particle

The height $h$ of a particle which is moving in space is given by the relation $$z=h(x,y)=\sin(xy+\pi)$$ The coordinates $(x,y)$ of the particle varies depending on the time $t:x(t)=e^t$ and we can ...
1
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0answers
38 views

Computing volume with triple integrals

I'm confused with this problem. Determine the volume of the solid limited for $x = 1-y$, $x = 3-y$, $y = 0$, $z = 0$ and $z = 1-y^2$. What I tried to do: well, first I suppose that the function I ...
0
votes
1answer
35 views

Complex-valued change of variables

If I have a function $f : \mathbb{R}^{2}\ni (x,y) \to f(x,y) \in\mathbb{C}$, I can define the change of variables $z = x + iy, \bar{z} = x - iy$, so that $f$ is now a function of $(z,\bar{z})$ and ...
0
votes
1answer
55 views

Determining the Euler-Lagrange equations for a minimizataion problem

I'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ where $\lambda$ is ...
1
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2answers
56 views

How to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis?

Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to ...
2
votes
1answer
57 views

Evaluating a double integral

I have to evaluate this double integral: $$\int_0^1\int_0^1\cos\ (\max \ \{x^3,y^{\frac{3}{2}} \} )\ dxdy$$ I have hint with me that this is to be done with help of Greens theorem but i dont know ...
1
vote
1answer
80 views

Deriving a high ordered Euler-Lagrange equation.

I've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the same ...
2
votes
1answer
41 views

What is an exact differential?

My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on ...
4
votes
2answers
102 views

Show Laplace operator is rotationally invariant

I'm trying to show the Laplace operator is rotationally invariant. Essentially this boils down to showing $$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 ...
1
vote
1answer
65 views

Gradient vs Conservative vector field: What's the difference?

From the definitions I'm reading between the two: The gradient vector field is defined by its construction: gradient of a scalar (or real) function generally over two or more variables. The ...
1
vote
0answers
41 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
3
votes
0answers
28 views

Multiple Integral Substitution Error

I just started learning about the substitution rule for multiple integrals and I decided to give myself an example problem: Calculate $\iint_R{(x^2 + y^2)dA}$ with $R = \{(x, y) \in \Bbb{R} \ |\ 0 ...
0
votes
1answer
81 views

one to one of multivariate function…

Suppose $f: \mathbb{R}^k \to \mathbb{R}^k $ has positive definite gradient matrix, $\dot{f}\equiv ( \partial f_i/\partial x_j )$. Then, where can I refer to see that $f$ is 1-1 or how to prove it? ...
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1answer
20 views

Double integral in cylindrical coordinates

I'm having trouble with a double integral problem in cylindrical coordinates. I'm sure the answer is staring me in the face, but I'm missing something. In the multivariable version of the Community ...
0
votes
0answers
15 views

Visualisation of gradient and computation?

I am learning differentiability in several variables, and I am stuck. I cannot visualize the definition that $\lim: \lim_{f(a+h)-f(a)-ch/h\to 0} = 0$ The book indicates that $ch$ is called gradient. ...
0
votes
1answer
65 views

chain rule with laplacian question

say I had a function $P(x,y)$ and I know that $\dfrac{\partial ^2 P}{\partial x^2} = - \dfrac{\partial^2 P}{\partial y^2}$ and wanted to show that $ P(x,-y)$ satisfied $ \Delta P(x,-y)$ (i.e. the ...
0
votes
1answer
49 views

Finding $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{\mathrm{d}z}{\mathrm{d}x}$

I'm given the following two relationships between $x, y, z$: $$\begin{cases} f(x,y,z)=x+y+z-1=0 \\ g(x,y,z)=x^2-2y^2+3z^2-2=0\end{cases} \quad .$$ Question: How would I go about calculating ...
0
votes
1answer
32 views

Solving coupled non-linear equations

I am struggling to understand what the following question requires me to do: I believe I need to differentiate implicitly, but am unsure how I show it cannot be done.
0
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1answer
207 views

Directional derivative - angle between the vector and coordinate axes

Doing an exercise a about directional derivatives, it was required to find the derivative of a given function $f(x,y,z)$ in the direction of the vector $ \vec{v}$ that forms with the coordinates axes ...
6
votes
2answers
84 views

Limit of 2 variables function

$$ \lim_{(x,y) \to (0,0)} \frac{\sin^2(xy)}{3x^2+2y^2} $$ If I pick $ x = 0$ I get: $$ \lim_{(x,y) \to (0,0)} \frac{0}{2y^2} = 0$$ So if the limit exists it must be $0$ Now for ${(x,y) \to (0,0)}$ ...
2
votes
0answers
46 views

Finding volume using triple integral

Find the volume of solid bounded by the $x^2+y^2=a^2$ , $y^2 + z^2 =a^2$ , $x^2 + z^2 = a^2$ I can see that shadow in $x$y region is given by $x^2 + y^2 =a^2$ . but when I draw ray from shadow to up ...
1
vote
1answer
35 views

Jacobian of composition of 2 functions.

Here is a question in my assignment which states that given: $$f(x,y)=(2x+y,3x+2y)~~~\text{and}~~~g(u,v)=(2u-v,3u+v)$$ , Find the jacobian matrix of $gof$ . I can't understand how to go with ...
1
vote
1answer
19 views

to find in which directions does Derivatives at a point exists?

If suppose I have a fuction e.g. $f(x,y,z)=|x+y+z|$ ,and I'm asked to prove that in which directions does derivative of $f$ at a point,(say $e_1-e_2$) it exists. How to think about the problem ?
1
vote
1answer
31 views

Exam question: Find the directional derivatives.

A question from a previous multivariable-calculus exam says: Let $D=\mathbb R^2$ and $$ f(x,y)= \begin{cases} 2xy/(x^2+y^2) & \text{if $(x,y)\neq (0,0)$,} \\ 1 & \text{if $(x,y) = (0,0)$.} ...