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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1answer
34 views

Sufficient conditions for differentiability of multivariate functions

Claim: If a function $f:\mathbb R^2\to\mathbb R$ has partial derivatives in a neighborhood $D$ of $(x_0,y_0)$, and if these are continuous at $(x_0,y_0)$, then $f$ is differentiable at $(x_0,y_0)$ ...
4
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1answer
45 views

How to show that $\varphi(x,y)=(x+f(y),f(x)+y)$ is bijective?

Let $f:\mathbb{R}\to\mathbb{R}$ be a $C^1$ function such that $|f'(t)|\leq k<1$ for all $t\in \mathbb{R}$. Let $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ be the function given by ...
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1answer
34 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
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0answers
40 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
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1answer
43 views

Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy $$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
1
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1answer
55 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
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0answers
28 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
2
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4answers
66 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
3
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2answers
143 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
0
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1answer
42 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...
2
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1answer
36 views

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
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2answers
36 views

solving double integrals

I'm trying to solve a double integral: $\displaystyle \int_{0}^{\frac{1}{2}}\int_{0}^{\frac{1}{2}-y}24xy\; dx \;dy$ I first solved in respect to $y$, making the $x$ a constant and plugged in the $y$ ...
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1answer
27 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
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1answer
40 views

Solid of revolution [closed]

Part A) Let T be a region in the first quadrant of the xz-plane bounded by the curves x 4 = z, x = 2, z = 16. Using the methods of 1-variable calculus, calculate the volume of the solid obtained by ...
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0answers
22 views

set notation for directional derivative: what does this comma mean?

Under the proof, for L,M what does the M stand for? Another function/set or domain? Also under the proof section, can someone give it to me in layman's terms why the first inequality is less than or ...
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1answer
22 views

Application of Implicit Function theorem for this problem

Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = \mathbf{0}$ and $$ Df(3,-1,2) = \begin{pmatrix} 1 & 2 & 1 \\ 1 ...
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3answers
80 views

Find the directional derivative of the scalar field

Find the directional derivative of the scalar field: $f(x,y,z)=\log(x^2+y^2+z^2)$ at $P_0(1,1,1)$ in the direction of the straight line $\ P_0P $ where $P=(3,2,1)$ What I have done: ...
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0answers
18 views

Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
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0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
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0answers
38 views

Lower boundary on hessian eigen values [duplicate]

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
1
vote
4answers
37 views

Help calculating the surface area given by the polar curve: $r=2(1-\cos\theta)$

I want to calculate the surface area given by the curve: $$ r = 2(1-\cos(\theta)) $$ using an integral. I have thought about doing this: $$ x = r\cos(\theta), \, y = r\sin(\theta) $$ $$ \iint r \,dr ...
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1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
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1answer
54 views

$f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $f(x,y)=(e^x \cos y,e^x \sin y)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $f(x,y)=(e^x \cos y,e^x \sin y)$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the ...
0
votes
2answers
34 views

Error in linearization: What is M in $| E(x,y) | \leq \frac{1}{2} M( |x-x_0 |+ |y-y_0 |)^2$?

I've been learning about linearization in multivariable calculus. $f(x,y) \approx L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$. and the error in this is $| E(x,y) | \leq \frac{1}{2} ...
3
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1answer
59 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
2
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1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
5
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1answer
49 views

Evaluating differential forms.

Can someone please check my work? It's an exercise from Barret O'Neill's Elementary Differential Geometry. I want to be really sure that my understanding of this is right. I see that the forms ...
0
votes
3answers
41 views

Triple integral calculation gone wrong?

Calculate the volume of the solid bounded $K$ by $$z \geqslant x^2 + y^2 - 1, \quad z \leqslant \sqrt{x^2+y^2} + 1, \quad z \geqslant 0$$ The triple integral I set up is $$ \iiint_K ...
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4answers
628 views

How to explain this quirk of the chain rule?

Assume I have a function $f = f(y, \phi(y,x))$ and I want to calculate $\frac{\partial f}{\partial y}$, I use the chain rule to get \begin{equation} \frac{\partial f}{\partial y} = \frac{\partial ...
0
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3answers
134 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
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2answers
47 views

double partial differentiation

I'm having troubles with solving problems with partial differentiations... and this one is double. I don't thing we've even learned this in class... Question: If $z=f(x,y)$, where $x=r\cos(\theta), ...
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1answer
31 views

partial differentiation of multivariable function

Let $a,b$ be constants satisfying $a^2+b^2=1.$ Let $f=f(x,y)$ be a twice differentiable function, and $u=ax-by$ and $v=bx+ay$. For the function $g$ defined by $g(x,y)=f(u,v)$, prove ...
4
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1answer
27 views

Computation of a certain flux integral

Let $$\Omega = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : \max(|x|_1, |x|_2, |x|_3) \leq 1\}$$ $$F_i(x) = \frac{x_i}{\|x\|^3}$$ and suppose $\varphi(y)$ be a continuously differentiable function of $y_i = ...
0
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2answers
47 views

If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
1
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1answer
66 views

Line Integral as Circulation - But why?

In most vector calculus texts say that if if the vector field $\vec{F}$ is viewed as the velocity vector of a fluid, then the surface integral $\iint_{S} \vec{F} \cdot d\vec{S}$, called flux, could be ...
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0answers
11 views

upper bound for error in multivariable differential

The question is, let $f(x,y,z)=x\cos(yz)$. by how much will the function $f$ change as the point $P(x,y,z)$ moves from $P_0(1,0,0)$ a distance of $0.1$ unit toward the point $P_1(1,1,1)$? Also derive ...
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1answer
15 views

Lagrange multipliers and angle between vectors

Can someone please help me with solving this question? I'm new to learning this and I'm not at all sure if what i've done is correct... The question is: the plane $4x-3y+8z=5$ intersects the cone ...
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0answers
28 views

Index of vector field $ \mathrm{grad} f (x)$ at critical point of index $\lambda$ is $(-1)^\lambda$

I don't understand why this is true. Near the critical point p, we have $f = -x_1^2 - \cdots -x_\lambda^2 + x_{\lambda +1}^2 \cdots x_n^2$, where $\lambda$ is the index of the critical point, and ...
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1answer
25 views

For a 2 variable function, prove that the linear approximation is less than the real value for all x and y

Consider $f : \Bbb R^{2} → \Bbb R$ defined by $f(x,y) = x^{2} + 3y^{4}$. Prove that $f(x,y) ≥ L(x,y)$ for all $(x, y)$ in $\Bbb R^{2}.$ I found the linear approximation: $$L = f(x,y) + f'x(a,b)(x-a) ...
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2answers
50 views

Multivariable calculus - scalar field

I don't know how to solve this problem. Determine if $\mathbf{F}$ is or not the gradient of a scalar field. If it is find the corresponding potential function f. $\mathbf{F}(x,y,z)= 3y^4 ...
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1answer
40 views

Calc 3 double integral

Compute the double integral of $f(x,y)=3\sin(5x)$ over the domain $D$ bounded by $x=0, x=\frac{\pi}{10}, y=0, y=\cos(5x)$. I am having trouble solving this double integral. I know that I must go ...
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1answer
39 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...
0
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0answers
24 views

Multivariate calculus “single” derivative as opposed to partial derivatives

say we have a function of two variables. f(x,y). One way to differentiate this is to split it up into two partial derivatives. For the partial derivatives you have to choose a direction (usually along ...
4
votes
2answers
50 views

limits using $ \epsilon - \delta $ to prove two variable function

I'm trying to use the $ \epsilon - \delta $ argument to prove $\lim_{(x,y) \rightarrow (1.1)} \frac{2xy}{x^2+y^2} =1$. I know that I need to show that $\forall \epsilon>0, \exists \delta>0$ ...
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2answers
26 views

Tips on resolving a Lagrange Multipliers equation system

I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance: The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 ...
2
votes
2answers
65 views

How to remember the Jacobian

the following is the problem that I was working on. Let $f(x,y)=8xy$ for $0<x<y<1$. What is the joint density function of $W={X \over Y}$ and $Z=Y$? Since I am self studying this ...
1
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0answers
34 views

Rearranging gradients in the chain rule

I wish to find the following \begin{align*} \frac{\partial f(x)}{\partial g(x)} \end{align*} where $x\in\mathbb{R}^n$, $f,g:\mathbb{R}^n\to\mathbb{R}$. Using the chain rule, with $y=g(x)$, as ...
0
votes
1answer
40 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
0
votes
1answer
24 views

Multivariable calculus problem on unitary disk

I came around this problem today: Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a differentiable map, constant on the disk $D=\left\{{(x,y): x^2+y^2=1} \right\}$. Prove that for every $X \in ...
0
votes
1answer
23 views

Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...