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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5
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1answer
45 views

Why curl of a vector field measures its tendency of rotation

I was trying to understand why curl measures a vector field's tendency of rotation. Two examples from physics seem to answer my question: Curl of the velocity field is twice the angular velocity ...
0
votes
2answers
43 views

Find a unit vector in which the directional derivative equals zero

Find a unit vector u in which the directional derivative of $f(x,y)=\ln(1-x^2-y^2)$ at ($\frac{1}{2} ,\frac{1}{2}$) is zero $f_x=\frac{-2x}{1-x^2-y^2}$ and $f_y=\frac{-2y}{1-x^2-y^2}$ Giving ...
3
votes
2answers
31 views

Continuity of a 2 variable function - Munkres exercise

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined as $f(0,0)=0$ and $f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2}$ for $(x,y)\neq (0,0)$ Then question asks to prove that $f$ is differentiable. Hint that ...
0
votes
2answers
37 views

Evaluate the double integral by changing to polar coordinates for $x^2+y^2\leq4$

Change the double integral $\int_{D}\int \sqrt{4-x^2-y^2}dxdy$ where D={$(x,y):x^2+y^2\leq4,y\geq0$} by changing to polar coordinates $r, \phi$ So am I right in thinking the limits would be 0 and ...
4
votes
3answers
47 views

Evaluate the integral using spherical coordinates

Given the integral $\int^{1}_{0}\int^{\sqrt{1-x^{2}}}_{0}\int^{\sqrt{1-x^{2}-y^{2}}}_{0} \dfrac{1}{x^{2}+y^{2}+z^{2}}dzdxdy$ I need to evaluate this using spherical coordinates. So far I have that ...
2
votes
3answers
51 views

Evaluate this integral using cylindrical coordinates

Find the volume of the solid bounded above by the paraboloid of revolution $z^{2}=x^{2}+y^{2}$ And below by the $xy$ plane, and on the sides by the cylinder $x^{2}+y^{2}=2ax$ We take $a>0$. ...
3
votes
1answer
15 views

Simple system of two nonhomogeneous ordinary differential equations solved by elimination. (3.1-15)

My differential equations textbook states to use the "elimination method" to crack this for $x$ and $y$. The final answer uses $t$ as the independent variable which both $x$ and $y$ are dependent on. ...
1
vote
2answers
46 views

The direction along which there is no change in value of $f=e^{x^2+xy}$ at point $(3,-2)$ is

The direction along which there is no change in value of $f=e^{x^2+xy}$ at point $(3,-2)$ is A. $(-0.6,-0.8) , (0.6,0.8)$ B. $(0.6,-0.8) , (-0.6,0.8)$ C. $(-0.6,-0.8) , (0.6,-0.8)$ D. $(0.6,0.8) ...
0
votes
1answer
94 views

To evaluate the integral $\int_{1}^{2}e^{x^2}\ln(x^2)dx$

To evaluate the integral $\int_{1}^{2}e^{x^2}\ln(x^2)dx$. I came across this while doing question $$\int_{1}^{4}\int_{\sqrt y}^{2} \frac{e^{x^2}}{y}dxdy.$$ I changed order of integration and now ...
0
votes
1answer
33 views

To evaluate multivariable limit $\lim _{(x,y)\to (0,0)} \frac{x^2y}{x^2-y^2}$

I have to evaluate this limit . $$\lim _{(x,y)\to (0,0)} \frac{x^2y}{x^2-y^2}$$ I think limit does not exists because when we approach from y=x, it blows up. which is not the case when we approach ...
0
votes
0answers
36 views

Continuous function on connected set is constant

I have asked one question previously differentiable map on a connected open set which says : $f$ is differentiable on $E$ and $E$ is open, conencted and $f'(x)=0$ for every $x\in E$ then $f$ is ...
0
votes
2answers
31 views

invertible function

for bijective function $f:U \to V$ , $f \in C^1$ $J_{D_{f}} \neq 0$ for any $ x\in U$ $, U\in \Bbb{R}^n, V\in \Bbb{R}^n $ open sets. I tried to show that $f^{-1} \in C^1$. I know the Inverse Function ...
0
votes
0answers
39 views

More intuition on the curl formula

I have a question regarding this quesiton. It says that $3$ simple fields that describe rotations around $x,y,z$ axis are: $$H_1(x,y,z)=(0,−z,y)\\ H_2(x,y,z)=(z,0,−x)\\ H_3(x,y,z)=(−y,x,0)$$ but why? ...
1
vote
1answer
72 views

Why is $F=\frac{-y}{x^2+y^2}\vec{i} + \frac{x}{x^2+y^2}\vec{j}, (x,y)\neq 0$ not conservative?

My book says that $$F=\frac{-y}{x^2+y^2}\vec{i} + \frac{x}{x^2+y^2}\vec{j}, (x,y)\neq 0$$ is not conservative (besides $curl(F)$ being $0$), so I cannot use the theorem that $$\int_\gamma \vec{F} ...
1
vote
1answer
27 views

Converting a parametric equation to non-parametric?

I am given a curve/path $\mathcal C$: $ x=2\cos u$ and $ z=2\cosh u$ where $ 0\le u \le \frac{\pi}{2} $ The surface of rotation $\mathcal A$ is given by rotating $\mathcal C$ around the Z-axis. Now ...
0
votes
0answers
21 views

Detail in the proof of formula for divergence

There is a common proof used to derive the formula $$\text{div}(F) = \dfrac{\partial F_x}{\partial x} + \dfrac{\partial F_y}{\partial y} + \dfrac{\partial F_z}{\partial z}$$ for the divergence of a ...
0
votes
0answers
47 views

How to simplify a problem with two variables?

I am trying to solve this problem. Let $\Delta$ be a positive number. I would like to find the values of $x$ and $y$ such that: $$ \left(1+\dfrac{x}{1+y}\right)\cdot\left(1+\dfrac{y}{1+x}\right) ...
0
votes
1answer
17 views

How to evaluate $\iint_S(x-y)^2\sin^2(x+y)dxdy$ , where $S$ is the parallelogram with vertices $(\pi,0);(2\pi,\pi);(\pi,2\pi);(0,\pi)$ ?

How to evaluate $\iint_S(x-y)^2\sin^2(x+y)dxdy$ , where $S$ is the parallelogram with vertices $(\pi,0);(2\pi,\pi);(\pi,2\pi);(0,\pi)$ ? Do I have to use a linear transformation of co-ordinates ? ...
0
votes
0answers
22 views

Verify my calculation of the surface integral without divergence theorem

I have $F=xyi-y^2j+zk$ Over surface $z=0$, $s \le1 $, $x^2+y^2 \le s$ My approach to calculate $ \iint F.ds$ was the outward normal is $k$ the dot product of this with F gives z so integral becomes ...
0
votes
0answers
34 views

Evaluating the integral $\int_{B_1(0)} \frac{|x|}{x^2+y^2+z^2} d(x, y, z)$

I was wondering if there is an easy way to evaluate the integral $$\int_{B_1(0)} \frac{|x|}{x^2+y^2+z^2} d(x, y, z)$$ where $B_1(0) = \{(x, y, z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 < 1\}$ is the ...
0
votes
2answers
23 views

Need to change variables in equations with cosh.

i have these five functions: $x=\tau \cosh(s)$ $q=\tau \sinh(s)$ $y= \sinh(s)$ $p= \cosh(s)$ $u= 1/2*\tau*\cosh(2s)+1/2*\tau$ I need to write $u$ in terms of $x$ and $y$ I know the answer is ...
2
votes
2answers
86 views

How to show $\DeclareMathOperator{curl}{curl}\curl\curl(e_r) = 0$

I want to figure out how to calculate $\text{curl}(e_r$). Where $e_r$ is a base vector for the Spherical co-ordinate system. Taking $e_r = (\sin\theta \cos\phi)i+(\sin\theta ...
1
vote
0answers
26 views

Application of stoke's theorem - Doubt on calculation of integral

This is an exercise question from Spivak's calculus on manifolds chapter number 4 question 26. Show that $\int_{C_{R,n}}d\theta=2\pi n$, and use stoke's theorem to conclude that $C_{R,n}\neq \partial ...
1
vote
1answer
22 views

Tangent Surface to a 4D Surface

I have been typing up notes for Multivariable Calculus. While doing so I have been pondering the terms I ought to use for higher dimensional surfaces and the associated tangent surfaces. With a curve ...
0
votes
2answers
22 views

Clarification of notation in multivariate taylor expansion

I'm reading the book "Numerical Optimization" by Nocedal and Wright and on page 14 of the book they present a form of the multivariate Taylor theorem which I find to be a bit peculiar. It is stated ...
0
votes
2answers
26 views

Exact ODEs and if $f_x\, dx + f_y \,dy = 0$ then $f(x,y)$ is a constant

In explaining a method to solve exact first-order linear differential equations, one uses the fact that if $f_x\,dx + f_y\,dy = 0$ then $f(x,y)$ is a constant. But the function $f(x,y)$ always turns ...
0
votes
0answers
52 views

$\mathbb R - \mathbb R$ notation

In multi-variable calculus I find this notation $\mathbb R - \mathbb R$ and $\mathbb R^m - \mathbb R^n$ the course material assumes that I am familiar with this notation. As an example "We usually ...
0
votes
1answer
13 views

Why Closed Area of Vector is conserved when Divergence of velocity vector is zero?

I had a line from some lectures saying that when div(v)=0, the area of the closed loop by the vector v is conserved on time. I really can't prove this or find the proof in online. Could anyone help ...
1
vote
0answers
30 views

Evaluate a derivative by definition

Consider the following function, where $f:\mathbb{R}^2\to\mathbb{R}\in C^1$: $$g(x,y) = y\cdot f(x^2-y^2)$$ So the partial derivative for $x$: $$g_x(x,y) = 2xyf'(x^2 - y^2)$$ Now, I want to get ...
2
votes
3answers
35 views

Change of variables from multiple to single

Consider the following limit calculation: $$ \lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} = \lim_{t\to 0} \frac{\sin t}{t} = 1$$ How can one justify this change of variables from multiple to ...
0
votes
1answer
14 views

how to show non-differentiability of function of two variables

I am currently reading Multivariate Calculus by Larson and Edwards. I understand the definition of differentiability of $z=f(x,y)$: if $\Delta z$ can be written as $$\Delta z = f_x\Delta x + f_y\Delta ...
0
votes
2answers
47 views

Example of a function with bounded second derivative but non-quadratic

Let $f:\Bbb R^d \to [0, \infty)$ be such that $f(x) \to \infty$ as $\|x\| \to \infty$, twice differentiable and $$|\frac{\partial f}{\partial x_i \partial x_j}| < C$$ for all $i.j$. I want to know ...
0
votes
0answers
17 views

first order contact iff their partial derivatives are equal

Let $f$ and $g$ be functions from $\mathbb R^n$ to $\mathbb R$ such that each has first-order partial derivatives defined everywhere on $\mathbb R^n$. Say that $f$ and $g$ have first-order contact at ...
0
votes
0answers
48 views

Directional derivate of gradient at point with given unit vector

Not entirely sure what I'm doing wrong here, here is the question, it is in Danish though, so I'll post a translation below: $$\mathbf f(x,y) = x^4 + 3x^3y^3 + 6y^2$$ Udregn den retningsafledede ...
0
votes
1answer
27 views

Evaluating the derivatives of implicit function

Given: $F(x,y,z)=x^2+3xy+2yz+y^2+z^2-11=0$. Does $F$ implicitly define a function $z = f(x,y)$ around the point $(x_0,y_0,z_0)=(1,2,0)$. If so determine ...
0
votes
2answers
38 views

total derivative with respect to x

I am trying to learn multi-variable calculus. I understand the meaning of partial derivative. But I am not getting the meaning of $$\dfrac{df(x,y)}{dx}$$. Please explain.
0
votes
1answer
18 views

Compute limit $\lim_{(x,y)\to(0,0)} \frac{|x|^{e^{1/|x|}}y}{x-y}$

Can someone help me compute the following limit (if it exists)? $$\lim_{(x,y)\to(0,0)} \frac{|x|^{e^{1/|x|}}y}{x-y}$$ Thanks in advance.
2
votes
1answer
30 views

The two corollaries of Stoke's theorem

The two corollaries of Stoke's theorem is as follows: 1) $\int \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}$ depends only on the boundary line, not on the particular surface used 2 $\oint ...
1
vote
0answers
20 views

Change of variables in a double integral - how to find the region??

I have the following question: Calculate the double integral $\int \int _D (x^4-y^4) e^{xy} dx dy $ on the region $D$, which is the set located in the first quadrant, bounded by the hyperbolas ...
0
votes
2answers
26 views

Calculating the volume bounded between a paraboloid and a plane

Will someone please help me with the following problem? Calculate the volume bounded between $z=x^2+y^2$ and $z=2x+3y+1$. As far as I understand, I need to switch to cylindrical coordinates: ...
0
votes
1answer
14 views

Determining the magnitude of forces acting on an object with no net force

Here is a screenshot of the problem: I understand we can write the vectors as say: $\mathbf{F_1} = \Vert \mathbf{F_1} \Vert \langle \cos\theta, \sin\theta \rangle$ I also know this problem ...
0
votes
0answers
26 views

General procedure for maximizing a function f(x,y) where x and y are restricted to stay along a known path in the xOy plane

I am looking for a general procedure for maximizing a function f(x,y) where x and y are restricted to stay along a known path in the xOy plane In the xOy plane there is a set of points $(x_n, y_n)$, ...
0
votes
0answers
20 views

What value of t do I choose for the following Directional Derivative

Find the directional derivative of $f\left( x,y,z\right) =x^{2}+yz$ At $\left( 1,3,2)\right)$ in the direction of increasing t along the path : $r\left( t\right) =t^{2}i+3tj+\left( ...
1
vote
1answer
31 views

$\int \delta(x + xy/u - a)\delta(y + xy/v - b)f(x,y)dxdy$?

I need help evaluating the following integral: $$\int \delta(x + uxy - a)\delta(y + vxy - b)p(x,y)dxdy$$ where $\delta(x)$ is Dirac-delta function, and $p(x,y)$ is some sufficiently well behaved ...
0
votes
1answer
12 views

Find maximal domain and range

$f\left( x,y\right) = \ln \left( 1-x^{2}-y^{2}\right)$ I have noticed that $1-x^{2}-y^{2}\gt 0$ But from here I am unsure Thanks
2
votes
1answer
42 views

Lagrange multipliers - regarding the theory and motivation

I am new with Lagrange multipliers , and having trouble understanding what is a necessary condition and what is sufficient. Assume I want to find global exterma of $f(x,y,z) \quad s.t. \quad ...
2
votes
4answers
39 views

Implicit Differentiation in multivariate calculus

Let $y(x)$ be the be given explicitly by the equation : $xy\left( x\right) -\ln y\left( x\right) = 1$ Determine $\dfrac {dy}{dx}$ I'm unsure of how to go about this problem.
1
vote
1answer
29 views

Multi-variable chain rule example

Let $p(x,y,z) = q(q(x^2, xy), q(xyz, sin(x^2y^2z^3)))$ where $q$ is a function of 2-variables. Find all partial derivatives. What I know/tried: Chain rule needs to be applied but the main issue is ...
0
votes
1answer
33 views

computing a triangular region using 'line integral' of stoke's theorem

Compute the line integral of the triangular region with vertices $\left ( 0,0 \right ),(2,0),\left ( 0,2 \right )$ with the function $\vec{v}=xy\hat{x}+\left ( 2yz \right )\hat{y}+\left ( 3xz \right ...
2
votes
3answers
50 views

There exists a non-empty open set $U ⊆ \Bbb R^2$ such that $f(x, y) = 0$ for every $(x, y) ∈ U$. Show that $f = 0$, i.e. $f$ is identically zero.

Let $f ∈ \Bbb R[x, y]$ be such that there exists a non-empty open set $U ⊆ \Bbb R^2$ such that $f(x, y) = 0$ for every $(x, y) ∈ U$. Show that $f = 0$, i.e. $f$ is identically zero. My try: Since $f ...