Questions tagged [multivariable-calculus]
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).
0
votes
0answers
6 views
Measure of $C^1$ path in $\mathbb{R}^2$
I started studying multivariable integration and still trying to grasp the conecpt of the measure.
I`m doing excersices and I keep getting the feeling im doing something wrong so I hope one of you ...
0
votes
4answers
28 views
Absolute maximum for $f(x,y) = 4x^2 + 5y$
Suppose you have a region $R$ that satisfies $4x^2+y^2\leq 4$ What is the absolute maximum of the function $f(x,y) = 4x^2 + 5y$ on $R$?
The correct answer is $10$
First, I calculated the critical ...
0
votes
1answer
11 views
Line Integrals of Vector Fields, Homework Conundrum
I am a student and I have a conflict with a given answer in the textbook. The question is the following:
Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for the given vector field $\...
0
votes
2answers
19 views
$\frac{\partial{z}}{\partial{y}}$ if $z$ is given implicitly
Suppose $z$ is given implicitly as:
$$e^z-x^2y-y^2z = 0$$
Find $$\frac{\partial{z}}{\partial{y}}$$.
I let $F(x,y) = e^z-x^2y-y^2z$. Then,
$$\frac{\partial{F}}{\partial{y}} = -x^2-2yz$$
$$\frac{\...
0
votes
1answer
24 views
Usage of Lagrange Multipliers in multivariable calculus
I just learnt about Lagrange Multipliers & am confused about why they are useful. Why can we not just check for critical points by checking if the gradient vector of the objective function $f$ is $...
1
vote
1answer
10 views
Compute mean and covariance matrix of $\bar{X}$ from a simple random sample
Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
0
votes
0answers
5 views
Deduce normal marginal distribution from joint distribution
Given two random variables $X$ and $Y$ with joint distribution
$F(x,y)=\phi (x) \phi (y) [1+ \alpha (1-\phi (x) )(1- \phi (y))]$
where $| \alpha | \leq 1$ and $\phi$ represents the standard normal ...
-1
votes
0answers
14 views
Divergence theorem for vector field over top half of sphere [on hold]
$$F= 2xz^2i + (2(y^3)/3 + \tan z)j + (2(x^2)z + 5y^2)k.$$ Evaluate $∫FdS$ over the surface $S$, where $S$ is the upper half of the sphere centered at the origin, with radius $1$, oriented upwards.
3
votes
0answers
32 views
Looking for another way to calculate the integral $\iint_{D}{\sin(x)e^{\sin(x)\sin(y)}}\text{d}A$
Here, I have a little unpleasant way to calculate the following double integral
$$\iint_{D}{\sin(x)e^{\sin(x)\sin(y)}}\text{d}A$$
where $D$ is the square area $D=\{(x,y)\in\mathbb{R}^2: 0 \le x \le \...
0
votes
0answers
17 views
How to prove that it is impossible to express one variable as function of others in a equation (implicit function theorem)?
Consider implicit function theorem. Since the conditions the theorem gives are only sufficient and not necessary I would like to know how can one prove that it is impossible to express (locally) one ...
1
vote
0answers
31 views
Intuition on Stokes' Theorem
Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int
\int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$
I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
1
vote
1answer
35 views
Show that there exists $T>0$ such that $\frac{e^{t}}{t^{1-\alpha}}-\int_{0}^{t}\frac{e^{\xi}}{\xi^{1-\alpha}}d\xi<0$ for all $t\geq T$
Let $\alpha\in\,]0,1[$. I want to show that there exists $T>0$ such that the positive function
$F(t):=e^{-t}\int_{0}^{t}\frac{e^{\xi}}{\xi^{1-\alpha}}d\xi$ decays monotonically to zero.
I ...
0
votes
1answer
30 views
The Jacobian of $(x,y)\mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.
I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the ...
0
votes
1answer
30 views
Surface Area of Multiple Integration
Find the area of the upper half of the cone $x^2+y^2=z^2$ above the interior of one loop of $r=cos(2\theta)$.
I know the formula for surface area is $\int_{x_0}^{x_1}\int_{y_0}^{y_1}\sqrt{(f_x)^2+(...
1
vote
0answers
27 views
Is $f(x,y) = \frac{x^ny^m}{x^{n'}+y^{m'}}$ continuous at origin? [duplicate]
Let function $\,\, f:\mathbb{R}^2\rightarrow\mathbb{R}\,$ be defined as
$$
f(x,y) =
\begin{cases}
\frac{x^ny^m}{x^{n'}+y^{m'}}\text{,} &\quad(x,y)\neq(0,0)\\
\text{0,} &\...
-1
votes
4answers
61 views
Does $\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$ exist? [on hold]
I am trying to solve the following
$$\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$$
This is basically $0/0$ form, but as I saw other post that L'Hôpital's rule does not work on multiple variables.
...
0
votes
0answers
12 views
Intervals of a Multivariable Function
If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
0
votes
0answers
33 views
Find $du$ and $d^{2}u$
Find $du$ and $d^2u$, if $u = f(xyz, xy^2, xz^2).$
For $du$, I did
$$du = f'_xdx + f'_ydy + f'_zdz$$ $$du =(f'_1yz+f'_2y^2+f'_3z^2)dx + (f'_1xz+f'_2*2xy)dy + (f'_1xy + f'_3*2xz)dz$$
How to proceed ...
0
votes
0answers
26 views
Derivative Of Infimum - Embedded Submanifold Of $\mathbb{R}^n$
This is a follow-up question to:
Derivative Of A Function Defined In Terms Of An Infimum
Let $A$ be a compact, (smooth) embedded submanifold of $\mathbb{R}^n$ and let $\Phi : \mathbb{R}^n \...
0
votes
1answer
31 views
Determine if a function is continuous in points
Determine if a function $f(x, y)$ is continuous in points (1, 1) and (0, 0).
$$f(x, y) = \begin{cases} \frac{x^2+2xy-3y^2}{x^3-y^3}, & x \neq y \\ A, &x=y=0 \\ B, & x =y= 1\end{cases}$$
I ...
2
votes
1answer
22 views
Changing variables or coordinates?
While posing another question I got stuck on the distinction of the following two concepts;
The components of a vector is usually referred to as it's coordinates, while trying to understand what "...
0
votes
1answer
26 views
How to study the critical points of a $2$-variable function?
I am revising some past exam questions and there is one that states:
Study the critical points of the function:
$$f(x,y)=x^2+y^2-x^4-y^4-2x^2y^2.$$
According to my professor, this is what I have ...
2
votes
0answers
34 views
Possible typing error in the book
One of the side effects of learning on your own is to seriously doubt if the book has an error or your reasoning is incorrect. My book on multivariable calculus (Rogawski 3rd Edition of Calculus ...
-2
votes
2answers
27 views
Limits for triple integral parabolic cylinder
Determine the volume bounded by the parabolic cylinder $z=x^2$ and the planes $y=0$ and $y+z=4$.
My work. I am not sure if I have the correct limits for this question. I used
$x = -\sqrt{z},\dots, \...
0
votes
0answers
7 views
No compact surfaces with constant causal character in $\mathbb{L}^3$
The title is all there is to it: I want to know why there are no compact surfaces with constant causal character in $\mathbb{L}^3$. If possible, some indications in the computing of the causal ...
2
votes
0answers
128 views
Total variation on a creased surface
Suppose $S$ is a smooth surface without boundary and $f:S\to\mathbb R$ is continuous. Then, we can define the total variation of $f$ as:
$$
\mathrm{TV}[f]:=\sup_{\|\phi\|_\infty\leq1} \int_S f(x)\...
3
votes
2answers
44 views
Triple integral of portion of cone (cylindrical polar coordinates)?
$V$ is the portion of the cone $z=\sqrt{x^2+y^2}\;$ for $\; x\geq 0.$
Find $$\iiint\limits_{V} xe^{-z} dV.$$
I am trying to solve this question. The answer is supposed to be just $4.$
I have worked ...
0
votes
0answers
10 views
Find the Fluid force on the vertical plate submerged?
Find the Fluid Force on a vertical plate submerged
B. Categorizing everything i see,
1.its "submerged in sea water" below the surface
2.the y-axis Depth = h(y) = -y = -6 ft
3.the x-axis at the ...
1
vote
2answers
41 views
Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.
Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$.
Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$.
really ...
5
votes
1answer
89 views
find closed form for following double integral containing radicals
question:
To prove :
$\displaystyle\int_{0}^{1}\displaystyle \int_{0}^{1}\dfrac{dxdy}{\sqrt{1-x^2}{\sqrt{1-y^2}}{\sqrt{4x^2+y^2-x^2y^2}}}=\dfrac{3\left(\Gamma{\dfrac{1}{3}}\right)^6}{2^{\dfrac{17}{3}...
0
votes
0answers
33 views
How to evaluate surface integral?
I got 4/3pi as my answer. But it is probably wrong because it only accepts an integer, fraction, or an exact decimal. I set up my integral as 2 on the very outside of my two integrals. I have (0 to ...
3
votes
4answers
83 views
Does this limit exist on $\mathbb R^2$
$(x,y) \in \mathbb R^2$
$$\lim_{(x,y)\to(1,1)} \frac{(x-y)^{(x-y)}} {(x-y)}$$
Does the limit above exist? Neither I could compute it nor I could find directions which have different limit values. ...
-1
votes
1answer
27 views
How can I calculate Fourier transform of this 3D function? [on hold]
Calculate Fourier transform of function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ defined by this formula
$$f(\mathbf{x})=\frac{1}{1+|\mathbf{x}|^2}.$$
0
votes
0answers
21 views
Integral of function undefined at one point
Let us consider a plane with polar coordinates. Let us also consider the following integral over any area $A$ on the plane:
$$\iint_A f(r,\theta)\ \hat{r}\ dr\ d\theta\ $$
Here the function is $\...
4
votes
2answers
42 views
When taking double and triple integrals over some region, how do I know what order to place the integral bounds?
Is there a general way of approaching this or does this vary on a case by case basis?
0
votes
0answers
18 views
Why can't I use gradient vector's magnitude as surface integral's area element?
I understand that when calculating the surface integral, we have to parametrize the surface by a 2-variables vector equation. Then, the cross product of the two velocity partial vectors would then ...
0
votes
2answers
32 views
double integration of floor function
I'm stuck on a double integration problem:
$$\iint ⌊x⌋*⌊y⌋ \,dA $$ over the region bounded by $x = -2$, $x = 1$, $y = 0$, $y = 2$.
I know that the region formed is a rectangle and the integral ...
0
votes
1answer
19 views
Differentiating an inner product w.r.t matrices
Let $M_n(\mathbb{R})$ denotes the space of all $n \times n$ matrices identified with the Euclidean space $\mathbb{R^{n^2}}$. Fix a column vector $x \neq 0$. Define $f:M_n(\mathbb{R}) \rightarrow\...
0
votes
0answers
10 views
Finding a bound for a sum over multi indices
I want to show that $\sum_{|\alpha||,|\beta|\leq N}sup\{t^{\alpha}D^{\beta}(x_1-t_1)_{+}^{N+1}\cdot\cdot\cdot(x_n-t_n)^{N+1}_{+}|t_i>0 \text{ for all i=1,...,n}\} $ is bounded by something in the ...
1
vote
1answer
33 views
Finding if a path is tangent to a level surface
The question is as followed:
Consider the path $r(t)=<t,2\sin2t,2\cos2t>$
TRUE/FALSE: the path $r(t)$ is tangent to the level surface $f(x,y,z)=x^2-y^2+z=1$
I guess what's throwing me off is ...
1
vote
1answer
27 views
Can I prove that a 2-variable limit does not exists if the limit on a curve is infinity?
Consider a 2 variable function $f(x,y)$ and the limit
$$\lim_{(x,y)\to (0,0)} f(x,y)$$
If I find two continuous functions $\gamma_1(t)$ and $\gamma_2(t)$ such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ ...
0
votes
1answer
25 views
Hydro Turbine Optimization
So I was doing the project related to Lagrange multiplier called hydroxyzine optimization recently and I have encountered problem on question4-6, which required me to plot a graph and find the optimum ...
-2
votes
1answer
30 views
How to differentiate $\int_{B(t)} f(x,t) dx$ with respect to $t$?
$\int_{B(t)} f(x,t) dx$ is given and assume $x$ is an $n$ dimensional vector variable, $t$ is positive real variable and $f(x,t)$ is a sufficiently smooth function of both $x$ and $t$. Also $B(t)$ is ...
-1
votes
0answers
29 views
Differential of a function with absolute values
I am trying to calculate the differential of the following function around $(0,0)$:
$$
f(x,y) = \begin{cases}
x, & |x| > |y| \\
y, & |x| \le |y|
\end{cases}
$$
I know that if $f$ is ...
0
votes
0answers
15 views
Calculating the differential and relating it to the directional derivative
I need to calculate the differential of the following function, but I am struggling to make sense of the relationship between the directional derivative and differential:
$f(x,y) = ln(x^2+2y+1) + \...
1
vote
1answer
60 views
Volume between cone and sphere of radius $\sqrt2$ with surface integral
Consider the cone $z^2=x^2+y^2$ between $z=0$ and $z=1$. Find the volume of the region above this cone and inside the sphere of radius $\sqrt2$ centered at the origin that encloses the cone.
The ...
-1
votes
1answer
24 views
How to change from a double to a single integral changing variables?
So I have the following integral:
$$I_1 = \iint u(x,y)dR$$
where $u(x,y)= e^{-(x^2 + y^2)}$
and the region $R$ is the rectangle $[-M,M]\times[-M,M]$.
I need to prove that $I_1$ equals:
$$I_2 = \...
0
votes
1answer
18 views
How do we conclude that $\nabla f(p) = 2np-\sum_{i=1}^n2p_i$?
I'm trying to show that $\bar{p}$ is a minimum
Writing the function as $$f(p) =\sum_{i=1}^n\langle p-p_i,p-p_i\rangle,$$we have $$Df(p)(v) =\sum_{i=1}^n 2\langle p-p_i,v\rangle.$$So, write this ...
1
vote
1answer
22 views
Partial derivative of a two variables function, one of which dependent on the other
I found this exercise on the book of multivariable calculus from which I'm studying:
"Find the partial derivative $\frac{\partial{z}}{\partial{x}}$ and the total derivative $\frac{\text{d}z}{\text{d}...
0
votes
0answers
18 views
Find the tangent space of Ellipsoid $M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$
Find the tangent space of
$$M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$$
So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (...
