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).

learn more… | top users | synonyms

1
vote
3answers
48 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 $(0,0)$?
5
votes
1answer
57 views

Compute $\iiint_V \sin^2 (x + y + z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is an ellipsoid.

By performing a suitable scaling and rotation of the coordinates, or otherwise, evaluate $\iiint_V \sin^2{(x + y + z)}\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is the region ...
1
vote
1answer
13 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
0
votes
0answers
8 views

Boundary curve C orientation when surface is oriented in the direction of the positive y-axis.

I am currently working through some Stoke's Theorem problems. One problem in particular, as described below, has been troubling me because the solution manual states that the parametrization of the ...
0
votes
0answers
20 views

Eliminating parameters to find cartesian equation?

I have vector equation $r(u.v)=(x+a_1u+b_1v)i+(y+a_2u+b_2v)j+(z+a_3u+b_3v)k $ How can I eliminate u and v to get cartesian equation?
1
vote
1answer
42 views

What are some good books on vector analysis in higher dimenesion

What is some good books specifically on vector analysis in higher dimension? Standard vector calculus book usually only introduced double and triple integral method
0
votes
1answer
17 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 ...
3
votes
0answers
39 views

Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...
1
vote
2answers
71 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
2
votes
1answer
28 views

Limit of a MultiVariable Equation

Hey guys I have a problem that asked to: Evaluate the following limit: $$\lim_{(x, y) \to (0, 0)} f(x,y) = \frac{2x^2y}{x^4+y^2}$$ along $y = mx$ and $y = mx^k$ I understand how to calculate the ...
4
votes
1answer
46 views

Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
0
votes
2answers
97 views

online software to draw generic graphics for calculus and analysis?

could you mention any software in which I could design images like this.
0
votes
1answer
63 views

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? [duplicate]

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? Using Langrange Multipliers, I've set up the standard equation with $$g(x,y) = (x/2)^2 + (y/3)^2 = 1$$ $$f(x,y) = ...
1
vote
1answer
24 views

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$. If the directional derivatives are continuous, does this mean $f$ is differentiable?

There is a result which states that for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ if all its partial derivatives exist and are continuous then the total derivative $Df$ exists. If I ...
2
votes
2answers
52 views

How do I prove that $\alpha/\lVert\alpha\rVert$ is differentiable?

Let $\alpha\colon I\rightarrow \mathbb{R}^3$ be a $C^2$-curve such that $\alpha(t)\neq 0$. How do I prove that $\alpha/\lVert\alpha\lVert$ is differentiable?
3
votes
0answers
41 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
0
votes
0answers
20 views

Question about the chain rule of matrix calculus?

From the wiki page matrix calculus, I learned the chain rule for matrix calculus is that (assuming numerator layout) ...
0
votes
4answers
37 views

Distance to origin from curve

Hello all I am trying to redo a problem I had and I am stumped for some reasons. I just want to find the maximum and minimal distance from the curve $$7x^2-6xy+7y^2-6=0$$ to the origin. But I want to ...
1
vote
2answers
28 views

Straight vs Partial derivative

Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
1
vote
1answer
42 views

Seeking Recommendation on Theoretical Multivariable Calculus textbooks

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I ...
0
votes
0answers
19 views

multivariable linearization

I have been asked to linearise the fallowing equilibrium points are phy=theta yaw=0 x,y,z=0 The idea I have using V'z as a model: -g+(kcm/m)(cos(phy)cos(thata)*voltages + ...
0
votes
0answers
33 views

Multivariable Chain Rule: Finding ∂z/∂y and ∂z/∂x

Question: The equation $$7xyz=2x^2+y^2+3z^2+7$$ implicitly defines z as a function of x and y in the neighborhood of the point where $x=2, y=1$ and $z=2$ . Find ∂z/∂x and ∂z/∂y at this point. ...
0
votes
0answers
14 views

Variant of Stokes theorem

I have recently, for the first time, seen the following version of Stokes' theorem stated: For a region $R$, $d\nu$ a volume element, $ds$ a surface element and $n(x)$ the unit normal to the surface ...
0
votes
0answers
24 views

Numerical Triple integral with three other parameters in R

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
0
votes
2answers
34 views

Integrating the derivative of a multivariate function

If $u=u(x,t)$ how do I compute $\displaystyle \int \frac{du}{dt}dt$ ? Would I be correct in saying it is not simply a case of cancelling the $dt$'s and getting an answer of $u$+ constant, seeing as ...
1
vote
2answers
43 views

Applying Chain rule to $z = z(u, v) = f(x(u, v), y(u, v))$.

If $z = z(u, v) = f(x(u, v), y(u, v))$ is a differentiable function, where $x = x(u, v) = u^2 − v^2$, $y = y(u, v) = 2uv$, show that $$\frac{∂^2f}{∂x^2} +\frac{∂^2f}{∂y^2} =\frac{1}{4(u^2 + ...
0
votes
2answers
58 views

Prove integrable function by partition

Let $f(x):[0,1] \rightarrow \mathbb R$, $$f(x)=\begin{cases} 1 & \text{ if } x=1/n ,\text{ $n$ is an integer }\\ 0 & \text{ otherwise }\end{cases}$$ a) Prove that $f$ is integrable b) Show ...
0
votes
0answers
10 views

Does the total derivative always require its inputs (e.g. $x,y$) to be defined parametrically (e.g. $x(r,\theta)$)?

Suppose we have $f(\theta,x(r,\theta),y(r,\theta))$ with appropriate conditions on $f$ such that we can take a total derivative. Then, the total derivative with respect to $\theta$ is $$\frac{\rm{d} ...
1
vote
3answers
20 views

Linear Approximation: Find the linear approximation at a point

Question: f(x,y)=√(7+2xy) Find the linear approximation at (3, -1) My answer: So I took the partial derivative at x and y, and I got Fx = y/(√(7+2xy)) and Fy = x/(√(7+2xy)). Evaluating for this, I ...
2
votes
1answer
23 views

Multi Variable Limit

Can anyone show me the steps? The limit is $0$ but I am facing some difficulties in getting to that point! I know that $\ln(1+u) \leq u$ for $u>-1$. $$\lim_{x,y\to ...
8
votes
2answers
83 views

Help to understand changing order of integration

I have a problem I have been working on, with the solution but the thing is I don't really understand how it is done. The question, is to compute, $$\int_0^1 \int_{9x^2}^9 x^3\sin(8y^3) \,dy\,dx $$ ...
0
votes
3answers
47 views

$\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, $k < l$. [closed]

How do I show that $\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, with $k < l$?
0
votes
1answer
32 views

using the divergence theorem

Use Gauss Divergence Theorem to compute $$\int \int \limits_S F\cdot n \,dS$$ where $n$ is the outward normal for the following: $S$ is the surface $x^2 +y^2=z^2$ and $z \in [0,1]$, and ...
0
votes
0answers
17 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
0
votes
1answer
14 views

differential of a product of two quantities

So far I've always blindly done a sort of product rule on a differential like this $$ d(\rho V) = \rho dV + V d\rho$$ I'm now wondering if it is also legitimate to write $$ d(\rho V) = \rho dV + V ...
-1
votes
0answers
20 views

Dam Break Problem

Is $u=0$, $h=1$ at every point where $x<-t$? $\frac{dx}{dt}=-1$ on $C_-$ and $\frac{dx}{dt}=1$ on $C_+$ only initially (at $t=0$) so how do we know that characterisitics intersect at every ...
0
votes
1answer
65 views

partial derivative of $\arctan ( \frac{x+y}{1+xy})$

partial derivative of $\arctan ( \frac{x+y}{1+xy})$ I am lost in finding the partial derivatives of the function. I started with the formula $\frac{1}{(1+x^2)}$. But it gets really complicated. Is ...
3
votes
3answers
92 views

The limit of $xy/(y-x^3)$ at $(0,0)$ does not exist

How to prove that $$\lim_{(x,y)\to (0,0)}\frac{xy}{y-x^3}$$ doesn't exist? Obviously, if $f(x,y)=\frac{xy}{y-x^3}$ and, for instance, $\gamma_1(t)=(t,0)$, we have $\lim_{t\to0}f(\gamma_1(t))=0$. We ...
0
votes
2answers
17 views

Finding limits of a volume triple integral in cylindrical coordinates

Find the volume between the cone $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=1$ that lies in the first octant (i.e., $x>0$, $y>0$, $z>0$) using cylindrical coordinates. It is obvious that ...
1
vote
2answers
25 views

Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$

Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$. I took partial derivative with respect to x, so $F_x = y^2 - 2xy + y$ $F_x = y(y - 2x + 1)$ Then with respect to y, $F_y = 2xy - ...
0
votes
0answers
19 views

Implicit solution to Method of Characterstics

If I have that, $\sqrt cu+v=$ constant along the characteristic lines $x+\sqrt c t=$ constant Where $u=u(x,t), v=v(x,t)$ and $c$ is constant. Why is it that $\sqrt cu+v=f(x+\sqrt ct)$ where $f$ is ...
1
vote
1answer
34 views

The method of Lagrange's Multipliers

I used the method of Lagrange's multpliers to find the maximum of $f(x,y,z)=\ln x+\ln y+3\ln z$ on the portion of the sphere $g(x,y,z)=x^2+y^2+z^2=5r^2 \ ; r>0$ where $x>0, y>0, z>0$ . I ...
0
votes
0answers
6 views

Taking divergence of a field?

Given $J = (z-hut)J_osin[B(\frac {Δz}{2} - |z|)]$. I want to find $∇.J$, its confusing because I don't really see any r, theta, or psi directions. I only see z-direction, which isn't r because r is a ...
0
votes
1answer
8 views

the diagonal angle between a theta and phi vector and the x axis as well as the derivative

Find w, the angle ∠rox. That is, the angle from the red vector r to the positive ...
1
vote
1answer
24 views

limits of r in cylindrical coordinates

Find the volume of a sphere $x^2+y^2+z^2\leq 1$ contained between planes $z=1/2$ and $z=1/\sqrt2$ using cylindrical coordinates. So the limits of $\theta$ would be $0$ to $2\pi$. Limits of $z$ would ...
1
vote
1answer
17 views

Two Definitions of Critical Points

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be given such that it is continuously differentiable. According to Wikipedia, a "critical point" of $f$ is a point $p\in\mathbb{R}^n$ such that: 1) According to ...
3
votes
1answer
30 views

Is the isosurface of a smooth function a smooth surface?

suppose $ f(x)\in C_c^{\infty}(R^n, R)$, an infinitely differentiable function with compact support, from $R^n$ to $R$. If $f\not\equiv 0$, is the boundary of its support, i.e. $\partial\{x\in R^n: ...
0
votes
0answers
18 views

Proof Check: For every partition $P$ of box $B$: $ \frac {F_+(B)}{V(B)} \leq \min_{c \in P}\frac {F_+(C)}{V(C)}$

I have to prove that for every partition $P$ of box $B$: $$\min_{c \in P}\frac {F_-(C)}{V(C)} \leq \frac {F_-(B)}{V(B)} \leq \frac {F_+(B)}{V(B)} \leq \max_{c \in P}\frac {F_+(C)}{V(C)}$$ Where V is ...
1
vote
0answers
27 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
8
votes
3answers
78 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...