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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0
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1answer
24 views

Calculate the line integral (cyl. coords)

So I have this vector field $$ \textbf{B}=K \left( \frac{\cos \varphi}{\rho^2}\textbf{e}_{\rho}+ \left( \frac{\sin \varphi}{\rho^2}+ \frac{1}{a\rho}\textbf{e}_{\varphi} \right) \right) $$ and the ...
0
votes
0answers
102 views

Centre of mass of the shape formed when a hemisphere is attached to a cone(an ice-cream cone shape)

What is the centre of mass of a hemisphere attached to a cone? I know that the bounds for the integration with respect to z must be the two equations( lower bound being the equation of the cone and ...
0
votes
2answers
172 views

Evaluate the integral by changing to spherical coordinates.

$$\int_{0}^{6} \int_0^{\sqrt{36-x^2}} \int_{\sqrt{x^2+y^2}}^\sqrt{72-x^2-y^2} xy~ dzdydx $$ I tried converting it and I ended up with ...
1
vote
3answers
63 views

How to compute $\lim_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} \frac{{{x^2} + {y^2} + {z^2}}}{{x + y + z}}$?

How do I solve the following limit: $$\mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} \frac{{{x^2} + {y^2} + {z^2}}}{{x + y + z}}$$
0
votes
0answers
14 views

General criterion for continuity and differentiability of $f(x,y)$

In a previous question I've posed, I've come to know that there is a criterion for check the continuity in the origin of a particular class of functions: $$ f(x,y) := \left\{ \begin{array}{rl} ...
0
votes
0answers
17 views

Multivariable functions estimation

Suppose that the directions of zero change of a function ๐‘“(๐‘ฅ, ๐‘ฆ, ๐‘ง) at the point (1,1,0) are ๐‘–โƒ— โˆ’ ๐‘—โƒ— and โˆ’๐‘–โƒ— + ๐‘—โƒ—. Suppose also that the derivative of the function ๐‘“(๐‘ฅ, ๐‘ฆ, ๐‘ง) increases ...
0
votes
1answer
42 views

Leading order Taylor Series Represention of the following function

I am given with this function $$f=\frac{1}{\sqrt{1+af_1(x)+bf_2(x)}},$$ where $$f_1=(1+x^2)^\nu,$$ and $$f_2=x^2(1+x^2)^{\nu-1},$$ where $\nu$ is a rational constant. I would want my $f$ to be of the ...
0
votes
1answer
17 views

Existence of partial derivative and their continuity in a neighbourhood of a point where it is differentiable

Let $f: \mathbb R^2 \to \mathbb R$ be differentiable at some $a \in \mathbb R^2$ , there is it necessarily true that there is a neighbourhood of $a$ in which all the partial derivatives of $f$ exists ...
0
votes
1answer
29 views

is $\frac{dx}{dz} + \frac{dy}{dz} = \frac{d(x+y)}{dz}$

asking this question few days ago, I first saw that "trick" in the given answer. that trick enabled me to solve the question pretty easily (not the solution given), by adding the 2 equations: ...
3
votes
3answers
74 views

On continuity and existence of all directional derivatives at a point of a scalar field whose gradient at that point is $\vec 0$

Let $f:\mathbb R^2 \to \mathbb R$ be a function such that for some $a \in \mathbb R^2$ , $\nabla f(a)$ exists and equals $\vec 0$. Is $f$ necessarily continuous at $a$ ? Do all directional ...
2
votes
1answer
30 views

Show a function is linear

I'm asked to show that for $x \in \mathbb{R^2}$ that $h:\mathbb{R} \to \mathbb{R}$ defined by $h(t) = f(tx)$ is differentiable by first showing it is linear. The solutions say to use $g(x) = ...
1
vote
0answers
10 views

$j$-fold summation

I need a fast way (closed form would be better) to calculate $ S=\sum_{a_{1}=1}^{a_{1}=c}\sum_{a_{2}=a_{1}}^{a_{2}=c}\dots\sum_{a_{j}=a_{j-1}}^{a_{j}=c}1$. I derived this from $S=\sum_{a_{1}\leq ...
0
votes
1answer
24 views

What is the slope of the intersection curve of the superfice $z=xe^{x^2y}$ with the plane $y=\ln2$ when $x=1$ [closed]

Help with this excercise hehe,, ok,, What is the slope of the intersection curve of the surface $$z=xe^{x^2y}$$ with the plane $$y=\ln2$$ at $x=1$?
1
vote
2answers
45 views

How do I use the Lagrange multiplier in this problem?

Suppose I have this problem: max f(x,y) s.t 2x + 4y = 3 If the function $f(x,y) = x^{0.7}y^{0.3}$ then how can I use Lagrange multiplier to find the ...
4
votes
1answer
61 views

If a function of two variables has a unique critical point, which is a local maximum, is it a global maximum?

$f(x,y)$ has partial derivatives in all $\mathbb R^2$ and a unique critical point at $(x_0,y_0)$ (local maximum). Is it a global maximum? I know that in compact sets, it isn't enough to say that ...
2
votes
2answers
39 views

f differentiable at (0,0) [duplicate]

Let $$f(x,y)=\left \{ \begin{array}{ll} \frac{x^2+y^2}{sin(\sqrt{x^2+y^2})} & \mbox{if } 0<||(x,y)||< \pi \\ 0 & \mbox{if } x=(0,0) \end{array} \right.$$ Determine if $f$ is ...
0
votes
0answers
13 views

Bijective relationship between two multi-variable functions

I have two functions, $f(x,y)$ and $F(x,y,z)$ I know the functional form of $F(x,y,z)$. But $f(x,y)$ is unknown. For instant, $$ F(x,y,z) = \frac{e^{-xz}}{ 1-e^{-xz}}xyz $$ In addition, we know that ...
1
vote
1answer
32 views

Show that $f$ is one-to-one and compute $f^{-1}$ explicitly.

Let $f = (f_l,f_2,f_3)$ be the vector-valued function defined (for every point $(x_1, x_2, x_3)$ in $\Bbb R^3$ for which $x_l + x_2 + x_3 \neq -1$) as follows: $$f_k(x_1, x_2, x_3) =\frac {x_k} {x_3 + ...
0
votes
0answers
34 views

Describe the curve $r=โŸจt,t^2,\cos(t)โŸฉ$

Describe the curve $r=โŸจt,t^2,\cos(t)โŸฉ$ I'm confused as to how I 'describe' the curve here?
1
vote
0answers
28 views

Integrability of subrectangle

A rectangle in $\mathbb{R}^n$ is a product of $n$ intervals $[a_1,b_1] \times [a_2, b_2] \times \dots \times [a_n, b_n]$. Let $Q = A \times B$ be a rectangle in $\mathbb{R}^{k+n}$, such that $A$ is a ...
1
vote
1answer
24 views

Use the fundamental theorem to evaluate the line integral?

For part a), I found $||r'(t)|| = \sqrt{18}(t^2+1)$ For part b), I solved $\int_{-1}^{1}f(t)*||r'(t)|| dt = \sqrt{2}\frac{24}{5}$ But I am unsure how to solve part c). I know that the path ...
1
vote
1answer
23 views

Reducing terms in the series expansion of a function of two variables

I have a function $f(x, y)$. This function is such that \begin{align} f(0, y)=a\\ f(x, 0)=a, \end{align} where $a$ is a constant. From this, a particular mathematician concludes: Thus if we ...
2
votes
1answer
45 views

Counterexample of linearity of the derivative

I found that the directional derviative $D_xf(0,0) = \sqrt{r^2+s^2} \cdot g \left ( \dfrac{r}{\sqrt{r^2+s^2}}, \dfrac{s}{\sqrt{r^2+s^2}} \right )$ for $x = (r,s)$ and I am then asked to show I ...
2
votes
1answer
55 views

Finding an equation of a plane perpindicular to xy plane that intersects with a surface and has a directional derivate of zero at this point.

I'm a bit new to 3D space and haven't had much practice with it. One question I'm working on says: A plane perpendicular to the x-y plane contains the point (3, 2, 2) on the paraboloid ...
-1
votes
2answers
34 views

Evaluating a Line Integral given r(t)

I've got a line integral to evaluate and I'm a bit lost as to how to work through it. The integral is given by $$ \int 3yz \, dx + z^3 \, dy + 3yz^2 \, dz $$ with the curve given by the ...
1
vote
0answers
50 views

does finding stationary points differ by method used to solve equations?

I am asked to find stationary points for the function: $z=x^3 + x^2 -xy + y^2 + 4 $ First step is to partially derive with respect to $x$ and then with respect to $y$ I get the following two equations ...
1
vote
1answer
17 views

Finding region for Change of Variables and Double integral problem

I'm running into some trouble on a problem in Vector Calc by Marsden and Tromba. I don't think I am correctly finding the region for my change of variables and the book doesn't have a similar example. ...
0
votes
1answer
23 views

Prove that $|$ det $g| v(U)$ is the volume of $g(U)$ for any linear transformation $g: \Bbb R^n \to \Bbb R^n$.

This is question of spivak's Calculus of Manifolds; (a) Let $g: \Bbb R^n \to \Bbb R^n$ be a linear transformation of one of the folยญlowing types : ๏ฟผ$$ \left\{ \begin{array}\\ g(e_i) = e_i, i\neq ...
1
vote
2answers
24 views

Interchange integrals in $\int_0^\pi\int_0^{\sin(x)}f(x,y)dydx$

I'm supposed to interchange the two integrals: $$\int_0^\pi\int_0^{\sin(x)}f(x,y)dydx$$ from dydx to dxdy Normally I'll draw the graph and integrate all the horizontal lines in function of $y$ and ...
1
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0answers
32 views

Proof of a certain inequality in two-dimensional Euclidean space

Please think it easy because it is not an assignment. I'm trying to show the following problem. Show that the inequality $$ ...
1
vote
1answer
28 views

Calculus/Analysis, show they are equal

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Prove that for each $x\geq 0$ $$\int_0^x \left(\int_0^t f(s)\,ds\right)\,dt=\int_0^x(x-s)\,f(s)\,ds$$ I think it's something related to change of ...
1
vote
0answers
22 views

Chain rule of multiple variables

I am having a hard time understanding the following: I need to have $\frac{dR}{dS}$, if I am given the following equations in terms of $x$ and $y$. $R(x,y)= x^3 + xy$ and $S(x,y)=6x^2y$. I was ...
0
votes
1answer
43 views

How do I solve this exact differential equation

The DE is $\;xy^4 dx + y^2 e^{-x} dy=0$. If I set $M(x,y) = xy^4$ and $N(x,y)=y^2e^{-x}$ then the $\int M(x,y) \; dx = \dfrac{(y^4x^2)}{2}$ and $\int N(x,y) \; dy = ...
0
votes
1answer
31 views

How to evaluate $\int_C y^2 dx + 2xydy$?

I am asked to evaluate the integral $\int_C y^2 ~dx + 2 x y~ dy$ where $c$ is the curve $(t^8,\sin^7(\frac{\pi t}{2}))$, where $0\leq t\leq 1$. My attempt so far is: $dx = 8t^7 dt$ and $dy = ...
0
votes
3answers
28 views

Finding the area of a parallelogram given vertices

Find the area of the parallelogram with vertices $(0,0)$, $(1,2)$, $(3,7)$, and $(2,5)$ I'm confused as to how I get started with this one. Can you all provide me with hints to assist me?
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0answers
33 views

Analyzing critical points when hessian determinant is null.

I'm doing this exercise in particular: $\begin{array}{*{20}{c}} {f(x,y) = {x^3} + {y^3} - 3x}\\ {\nabla f(x,y) = (3{x^2} - 3,3{y^2})}\\ {\left\{ {\begin{array}{*{20}{c}} {3{x^2} - 3 = 0}\\ {3{y^2} = ...
1
vote
0answers
33 views

Application of Gauss divergence theorem or Grren's formula in n dimension

Let $B_\rho$ denotes the ball of radius $\rho$ and center at $0$ in $\mathbb{R}^n$ then prove the following: $$\int_{B_\rho}u_{x_i}dx=\int_{\partial B_\rho} u \cos\langle r,x_i\rangle dS $$. I know ...
1
vote
3answers
49 views

How to evaluate this line integral?

I am asked to evaluate the line integral $\int_C x^2dx + xy^2dy + dz$, where $C: [0,1] ->R^3$ is given by $c(t) = (x(t),y(t),z(t)) = (t^2,t,1)$ I started to approach this question by saying that ...
0
votes
0answers
14 views

Partial derivative of a definite integral whose bounds are variables

Suppose I have $f(a, b)$ such that $$ f(a,b)=\lambda\int^b_ah(\tau)d\tau. $$ What is $\frac{df}{da}$, and how is it obtained? My work We have \begin{align} ...
1
vote
2answers
45 views

does there exist nondifferentiable $f$ that is continuous AND has all directional derivatives $D_u$ AND $D_u=\nabla f\cdot u$?

There are many standard examples of functions $f:\mathbb{R}^2\to\mathbb{R}$ that possess all directional derivatives at a point and yet fail to differentiable or even continuous there. The most ...
0
votes
0answers
12 views

Incomplete answers with chain rule and directional derivatives

In this question asking to use the chaing rule to compute dz/dx & dz/dy, I always seem to get a missing 1/2. Where does it come from? $$ 2x^2+3y^2-2z^2 = 9 $$ I start with separating z. $$ 2z^2 ...
0
votes
0answers
11 views

Under what conditions is $|Du|=|\frac{\partial u}{\partial \nu}|$ at a point?

I was reading another person's solution to an exercise here, and have trouble understanding how their last line proves the claim. Certainly by CS we have $|\frac{\partial u}{\partial \nu}|=|Du \cdot v ...
0
votes
0answers
21 views

Evaluating line integral

I have this problem: A uniform wire has the shape of that portion of the curve of intersection of the two surfaces $ x^2 + y^2 = z^2 $ and $ y^2 = x $ conecting the points $ (0,0,0) $ and $ ...
0
votes
3answers
4k views

How to prove this vector identity [closed]

How do i prove this vector identity ? $$(\vec a \times \vec b)\times \vec c=(\vec a \cdot\vec c)\vec b - (\vec b\cdot\vec c)\vec a$$
1
vote
1answer
37 views

Using Lagrange multipliers to find max and min values?

First off, I'd like to apologize, but I'm unfamiliar with how to format math equations on this platform and am not finding anything to fix that. I've been given the problem: Use Lagrange multipliers ...
0
votes
1answer
49 views

Triple Integral with a tetrahedron as the domain (given vertices of the tetrahedron) [closed]

Evaluate the triple integral: $$\iiint_Txz \, dV$$ where $T$ is the solid tetrahedron with vertices at $(0, 0, 0)$, $(1, 0, 1)$, $(0, 1, 1)$, and $(0, 0, 1)$. The answer that I calculated was ...
0
votes
1answer
56 views

Characterizing the critical point of a two-variable function when the Hessian determinant is zero

I want to find the critical points of the function $$f(x,y)=x^4+y^4-y^2.$$ So, I found the partial derivatives and I set them to zero and solved for $x$ and $y$. I ended up with $2$ points: $(0,0)$ ...
0
votes
0answers
27 views

Use the Lagrange multiplier method to compute the maximum value of the function?

Use the Lagrange multiplier method to compute the maximum value of the function $h(x,y,z) = x+z$ on the sphere $x^2+y^2+z^2=1$ My attempt: โˆ‡h = (1,0,1) = ฮป(2x,2y,2z) This implies x = z and y = 0 ...
1
vote
1answer
25 views

Use Sylvestor's Criterion to classify the critical point?

I am asked to find and classify the critical point of $g(x,y,z) = xy-x+2z-x^2-y^2-z^2$ I have calculated the derivative and found that the critical point is at $(-\frac23, -\frac13, 1)$ I have found ...
0
votes
0answers
21 views

How to calculate this integral $\int_M f$?

Let $M= \{(x,y,z):x+y^2=0, z^2-y^2=0, y^2>0\}$ and $f(x,y,z):= \left\{\begin{array}{cl} (z-1)(3-z), & \mbox{if } 1\leq z\leq 3\\ 0, & \mbox{otherwise} \end{array}\right. $ I have to ...