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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4
votes
0answers
21 views

Volume between sphere and cylinder with different centers

I am working on a tumor model and need to calculate the volume enclosed between the sphere given by $$(x-d)^2+y^2+z^2=r^2$$ and the cylinder given by $$x^2+y^2=R^2.$$ I have worked it out by using ...
0
votes
0answers
9 views

Leibniz rule under double intergral

I have looked through other posts about Leibniz rule under integral, and I have done with my problem. But I am not sure if it's correct or not, please check it for me if you don't mind. ...
-2
votes
0answers
7 views

Writing a multi-variable function [on hold]

An oil well needs to be drilled to reach a reserve 3miles below the surface. Unfortunately, the closest we can get along the surface is 10miles away. Between the surface and the reserve, there are two ...
1
vote
2answers
23 views

Finding the absolute maximum of the following 3d function

$ f(x,y) = \frac{(\lambda_1x+\lambda_2y+\lambda_3)^2}{x^2+y^2+1} $ I know that the function looks like some deformed dorito chip depending on the lambda values. That is about as far as I've gotten. ...
0
votes
0answers
13 views

Tangent line to the intersection curve

The point $P(1,2,3)$ is on the intersection curve of the two surfaces $f(x,y,z) = {y^2} + \ln (xz - y) - {x^2}z - 1 = 0$ and $g(x,y,z) = \sin (x + y - z) + z \cdot {e^{y - 2x}} - 3 = 0$. At ...
1
vote
0answers
20 views

Why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$?

In this image, why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$ ? [I borrowed link to the image uploaded by ...
0
votes
0answers
14 views

Compute the work done by the force field →F (x, y) = (tan(x) +yx2 ) →i + (x 2 −y 3 ) →j in moving a particle counterclockwise around the unit circle.

I am unsure of the parametrization of this. As it is the unit circle I was thinking →r(t)=(cos(t))→i + (sin(t))→j However when I did that, I became very stuck on the integration leading me to think ...
2
votes
1answer
26 views

Cartesian into polar integral.

I have set up an double integral to prove gauss theorem in physics for a gaussian surface of cube of edge $a$ which is as follow. I supposed that mid point of cube is at origin and a charge is placed ...
-5
votes
1answer
26 views

Numbers with variable power is poitive. [on hold]

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a<0$.
2
votes
0answers
39 views

Div, grad, curl in curvilinear coordinates

I've a lot of different formulas for div, grad, curl, and laplacian in different coordinate systems. How are these formulas derived? What's the general procedure for finding the formula of say the ...
-2
votes
2answers
39 views

Inequality of numbers.

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a\le0$. (May be Jensen's inequality help but need help how to apply.)
1
vote
0answers
20 views

Derive chain rule for complex functions, from the chain rule for real functions

I'm trying to obtain the chain rule for complex (not necessarily holomorphic) functions $\mathbb{C} \to \mathbb{C}$, using the known chain rule for functions $\mathbb{R}^2 \to \mathbb{R}^2$. The ...
1
vote
1answer
14 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
0
votes
3answers
75 views

Graph connected does not imply $f$ is continuous [on hold]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
0
votes
2answers
45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
0
votes
0answers
17 views

Problem with center of mass in polar coordinates

When we calculate center of mass using rectangular coordinates, we find the average values in each coordinate. Obviously we can't do this very same thing in polar coordinates: if we integrated a ...
0
votes
2answers
23 views

Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
0
votes
0answers
18 views

Correctly setting up flux integrals

My question has to do with picking the correct limits for integration. I thought I had it figured out well, but I had an interesting issue with a homework problem. The problems were about Green's ...
0
votes
0answers
22 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
0
votes
1answer
26 views

Integrability of characteristic function

I have a following questions that I am having trouble. Let $E = \{(\frac{a}{b}, \frac{c}{b}) : a,b,c \in \mathbb{Z}, a \text{ and } b \text{ are relatively prime}\}.$ For what $a \in [1,2]$ is the ...
0
votes
1answer
28 views

Which subset of $\mathbb{R}^n$ has zero content

I have a question related to zero content. The questions are For what $n$ there an open subset of $\mathbb{R}^n$ that has zero content? For what $n$ there an unbounded subset of $\mathbb{R}^n$ that ...
1
vote
1answer
46 views

Gradient of a vector function

I have a vectorial function $f$, defined on the set of all $n$-dimensional vectors. $f(x) = \log(x^TAx)$, where $\log$ is the natural logarithm, $x^T$ is $x$ transpose and $A$ is a symmetric $n \times ...
0
votes
0answers
17 views

Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
0
votes
0answers
20 views

Change of variables in double integration

I was trying to solve this double integral $\int_{0}^{1}\int_{0}^{y}(1-x)^{59}(y-x)^{27}dxdy$, I could do this by taking binomial expansion but that would be very painful. So a sure thing here is a ...
2
votes
1answer
14 views

Finding surface area S using area of projection of S??

I was going over my calculus textbook and came across a question about surface area. and question is as follows. Let S be a parallelogram not parallel to any of the coordinate planes. Let ...
1
vote
0answers
63 views

Calculating electric flux through a sphere (calculus)

This problem comes with two parts, and the reason I am posting here is that they are both supposed to result in the same answer but I am getting two different values. A spherical shell of radius $R$ ...
-1
votes
2answers
37 views

Volume of solid by Spherical

Trouble setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. Use integration with Spherical coordinates. (Hint: Use two ...
0
votes
0answers
12 views

Rotation Matrix in domain and co-domain basis

I was asked t o derive the rotation matrix counterclockwise with given angle in different domain and co-domain basis. Using what we know from trigonometry I derived the Rotation matrix as: R(Q) = ...
1
vote
1answer
11 views

How to find vector $\vec{A}+\vec{B}$ with position vector and displacement vector using different methods

When position vector $\vec{A}$ is $\langle 4, -2, 3\rangle$ and displacement vector $\vec{B}$ is from point $Q(0,4,1)$ to point $R(2,3,-2)$ How am I supposed to find vector $\vec{A}+\vec{B}$ using ...
1
vote
0answers
8 views

Does a function need to have a parameter underlying all inputs to be totally differentiable?

I have the following function $$e(p_x,p_y,\bar{U}) = p_x a(p_x,p_y,\bar{U}) + p_y b(p_x,p_y,\bar{U})$$ Can I take the total derivative of this? I am confused about the criteria for something to be ...
2
votes
2answers
47 views

Having trouble calculating $f_{xx}$ of a “variable-heavy” quotient.

Let $$ f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $$ Compute $f_x (0,0)$, $f_y (0,0)$, $f_{xx} (0,0)$, $f_{xy} (0,0)$, and ...
1
vote
3answers
44 views

How do I prove that the limit of $\frac{x^2 y }{x^2 + y^2} = 0$?

How do I prove that $\lim_{(x,y)\to (0,0)} \frac{x^2 y }{x^2 + y^2} = 0$? I can prove this by notifying $x=rcos\theta$ and $y=rsin\theta$, but I remember that it could also be proven by squeeze ...
1
vote
0answers
74 views
+50

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. Let $f:X\to Y$ and $a\in X$ There is a map $\Delta : X \to L(X,Y)$ continuous at ...
0
votes
2answers
59 views

Line integral of conservative vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
1
vote
1answer
37 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
0
votes
1answer
31 views

Double integral variable change help

I'm having a tough go with this problem. $\iint \frac{x^2}{y^3} dA$ , Integrate using a change of variables over the region defined inside the curves $y=2x,\; y=x,\; y=x^2,\; y=2x^2$ . I graphed it ...
1
vote
1answer
33 views

Double Integral Change of variable help

I am having some trouble getting this problem set up, and would appreciate any help. Problem: $\iint \frac{1}{(x+y)^2} dA$. Integrate using change of variables over the region inside the lines ...
1
vote
1answer
24 views

How do I simplify a Multivariable expression involving derivatives of logarithms?

I have this expression I got after a lot of calculation: $$\sigma =\frac{d\log\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)}{d\log\left(\frac{ 2 ...
0
votes
3answers
35 views

Why would the Jacobian not be zero in this case?

Find the jacobian of the transformation x = u, y = 3uv in the uv plane. Why would $U_y$ not be zero in this case, if the equation U = x contains no mentions of y?
2
votes
1answer
26 views

Green's Theorem and limits on y for flux

I'm working through understanding the example provided in the book for the divergence integral. The theorem (Green's): $$ \oint_C = \mathbf{F}\cdot \mathbf{T}ds = ...
2
votes
3answers
64 views

Question about maximizers and trig

Hi there I have a quick question about the following Consider the simple maximization problem of $$f(x,y)= \frac{x}{1+x^2+y^2}$$ It can be easily seen from analysis of critical points obtained from ...
0
votes
1answer
19 views

Two variables Taylor's expansion

I guess that Taylor's expansion about $(0,0)$ is useful for finding value of $\dfrac{\partial^{4n}}{\partial x^{2n}\partial y^{2n}} \left (\dfrac{1}{1+x^2+y^2}\right)(0,0) $. How can it do?
0
votes
1answer
41 views

What does it mean for partial derivative to be continuous and how does that imply differentiability?

In order for function to be differentiable at some point, it should be well approximated at that point. I understand that partial derivatives must exist, and that function needs to be continuous, but ...
0
votes
0answers
11 views

Taylor's expansion of vector valued functions

Apostol, Calculus, Vol 1 : Let $b$ be a given point in $\mathbb R^n$. Then, if $v$ be any given vector,my textbook defines the taylors expansion of $f$ as : $f(b+v) = f(b) + f~'(b)(v)+ ||~v~|| ...
2
votes
1answer
38 views

Riemann Integral on $\mathbb{R}^2$

I have the following question. Find a function $f(x,y)$ that is integrable on rectangle $[0,1] \times [0,1]$, such that $g(y) = f(\frac{1}{2}, y)$ is not integrable for $y \in [0,1]$, or prove that ...
2
votes
1answer
27 views

Level surface undefined

Can a level surface be undefined at some point, even if the original fuction is defined at the same point? example: $w(x,y,z) = xy+yz+xz$ is defined at $p=(1,-1,2).$ Its level surface at $p$ is ...
0
votes
1answer
20 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
1
vote
3answers
44 views

I need hints to proof $\lim\limits_{(x,y)\to(x_0,y_0)} \frac{\sin f(x,y)}{f(x,y)} = 1 $

Let $f: U\subseteq\mathbb{R^2}\to\mathbb{R}$, $(x_0,y_0)\in U$ and suppose that $\lim\limits_{(x,y)\to(x_0,y_0)} f(x,y) = 0$. Prove that $$\lim\limits_{(x,y)\to(x_0,y_0)} \frac{\sin f(x,y)}{f(x,y)} = ...
3
votes
2answers
26 views

parametric equation of level curve in three dimensional plane

What is the parametric equation for the tangent plane to the level curve of the function $$w(x,y,z) = xy+yz+xz$$ at the point $(1,-1,2)$? My answer was: $$(x,y,z) = ...
0
votes
2answers
31 views

Different results for the same equation

Why does the chart of $xy+yz+xz=-1$, a one sheeted hyperbolid, is different from the chart of $z = -\frac{1}{x+y} - \frac{xy}{x+y}$? Aren't they both the same equation?