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
0answers
26 views

Drawing for multivariable calculus.

Does anyone have any step-by-step instructions for how they typically go about drawing 3d images, slices, and projections. I'm having a really hard time looking for good outlets/examples on how to ...
0
votes
1answer
150 views

Can anyone help me with this double anti-derivative? None of my teachers can…

The integral is the following: $$\iint\sqrt{r^2-x^2-z^2}dxdz$$ Where r is a constant and x and y are both variables. I have filled pages and pages trying to solve it, i think it can be solved by ...
1
vote
1answer
72 views

Finding Extreme Values (Multivariable)

Given $f(x,y)=x^2+2y^2$, find its extreme values on $x^2+y^2=1$. I know how to solve this problem using Lagrange's method and the constant variation method. The solutions are $(\pm1,0)$ and $(0,\pm1)...
0
votes
1answer
31 views

Multivariable calculus & partial derivatives problem

Let $$H = \mathbb f(S,V) $$ $$\dfrac{\partial H}{\partial S}S\sqrt{V} = \mathbb g(H) $$ $$\dfrac{\partial H}{\partial V}\sqrt{V} = \mathbb h(H) $$ Note that functions $\mathbb g$ and $\mathbb h$ ...
2
votes
3answers
81 views

Continuity of a function $f: \mathbb{R}^2 \to \mathbb{R}$

It's easy to check that the function $$ f_1(x, y) = \begin{cases}\frac{x y}{x^2 + y^2} &\text{if (x, y) ≠ (0, 0)}\\0&\text{if (x, y) = (0, 0)}\end{cases}$$ is not continuous in $0$, because ...
0
votes
0answers
44 views

Partial derivatives and chain rule explanation.

I have a function $w=f(x,y)$, where $x=r\cos{\theta}$ and $y=r\sin{\theta}$ and I'm asked to show that $$\frac{\partial w}{\partial x}=\frac{\partial w}{\partial{r}}\cos{\theta}-\frac{\partial{w}}{\...
0
votes
2answers
162 views

What's the parametric equation for the plane through a point (x,y,z) perpendicular to (a,b,c)?

Find the parametric vector and Cartesian equations for the following planes: a. The plane thru point $(2,1,-2)$ perpendicular to vector $(-1,1,2)$. b. The plane thru the three points $(2,2,-2)$, $(-...
1
vote
0answers
81 views

Linear Algebra requirement Spivak's Calculus on Manifolds

I am interested in the extent of knowledge of Linear Algebra required for Spivak's Calculus on Manifolds. More precisely, in the first problems in his book they reference norm preservation and inner-...
1
vote
1answer
435 views

Determine if first and second partial derivatives are positive, negative or zero based on level curves

Assuming I have a point on a level curves graph for function f(x,y), how would I determine whether the first and second partial derivatives are positive, negative, or zero? I understand that for a ...
1
vote
0answers
114 views

Find the area portion of Surface of $x^2+y^2+z^2 =b^2$ inside a cylinder

Find the area of the surface portion: $x^2+y^2+z^2=b^2$ that remains within the right cylinder $ x^2+y^2=by$ , where $b > 0$
0
votes
1answer
57 views

Is this a sufficient condition for differentiability

Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that, for all $c\in \mathbb{R}$, every vector in $f^{-1}(c)$ is supported by a unique hyperplane to $f^{-1}(c)$. Is $f$ ...
2
votes
4answers
69 views

What is the physical meaning of this integral?

Let $$I=\int_S ~z~dS$$ Where $S$ is the surface of a hemisphere with equation $x^2+y^2+z^2=4~~~~z \geq0$. I know $$\int_S~dS$$ would be the surface area of the hemisphere but I can't figure out how ...
3
votes
2answers
55 views

Differential equations: IVP application

I am given $$y'=0.05y-800$$ I am asked to: (a) Find all constant solutions of the differential equation. (b) Suppose $y = M$ is your constant solution from (a). Plot two solutions of the ...
2
votes
1answer
161 views

Give a formula for the volume of the solid under a surface $z=xy$ and a triangle?

Given is the solid with unit density lying under the surface $z = xy$ and above the triangle in the $xy$-plane with vertices $(0, 1, 0)$, $(1, 1, 0)$ and $(0, 2, 0)$. Give a formula for the ...
5
votes
1answer
48 views

$x_t := a_t -b_t c_t $ , with $dx_t = \theta (\mu-x_t) dt+ \sigma dW_t$

I would like to solve the following equation explicitly using Ito's lemma: $$ x_t := a_t -b_t c_t , $$ where $x_t$ is an Ornstein-Uhlenbeck process (see here) $$ dx_t = \theta (\mu-x_t) dt+ \sigma ...
1
vote
0answers
22 views

Inverse of a Bijective Bivariate quadratic function or polynomial

I am looking for some general way to invert a bijective quadratic polynomial of the form $$ f(x,y)=A_0x+A_1x^2+Axy+B_0y+B_1y^2+Byx $$ where the coefficients may or may not be in the same ring as the ...
3
votes
3answers
41 views

Find function $f: U \subset \mathbb{R}^2 \to \mathbb{R}$ with $||Df(x,y)|| \leq 1$ such that $P,Q \in U$ exist with $|f(P) - f(Q)| > || P-Q||$

I am looking for a continuously-differentiable function $f: U \to \mathbb{R}, U \subset \mathbb{R}^2$ which satisfies the following requirements: $U$ is open set $U$ is connected (right word?), i.e. ...
1
vote
2answers
615 views

Proof that a log-of-sum-of-exponentials is a convex function

It's well known in statistical mechanics that the following is a convex function of the vector $\theta$: $$ A(\theta) = \log \left( \sum_{i=1}^\infty e^{\theta \cdot f(i)} \right) $$ where $f(i)$ is ...
1
vote
1answer
53 views

Lagrange multipliers problem with two constraints

Hi guys I am working with the following polynomial and I am trying to find the $\lambda , \mu$. I have a polynomial and I am trying to do Lagrange multipliers. Here is what I have. $f(x,y,z)= a \log(...
2
votes
1answer
29 views

Find Maxima and Minima

Sir, help me to find the maxima, minima & saddle point. $$f(x,y)=x^2-x^2y+2xy^2+4y$$ here $$\begin{align}f_x(x,y) & =2x-2xy+2y^2 \\ f_{xx}(x,y) & =2-2y \\ f_y(x,y) & =-x^2+...
2
votes
2answers
52 views

Multivariable Calculus -Jacobian of the transformation

I am trying to figure out the answer to this problem: Evaluate $$\iint sin(\frac{x+y}{2}) cos(\frac{x-y}{2})dA$$ on $R$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. I ...
0
votes
1answer
46 views

Double integrals: How to choose the order of the limits of integration

I have to calculate the integral of $|x|$ in the region that you see here: I have some doubts about the point from which start to integrate and about the order that I have to follow for the limits ...
-1
votes
1answer
81 views

Double integration of $\frac{1}{\sqrt{x^2 + y^4}}$

I am just learning double integration. I am stuck with the following problem: $$\int_{\mathbb{R}^2}\frac{1}{\sqrt{x^2 + y^4}}\,dx\,dy$$ I am not even sure whether is integral is finite. I would ...
-4
votes
2answers
59 views

numerical techniques of integration.

I am working on some past exam questions of integration and i came across this question. Can any body solve & explain this in detail to me. Thanks using FTC we Know $$\int_1^3 \frac{1}{x^2} = \...
1
vote
2answers
49 views

Differentiability of $f(x,y)=2xy+\frac{x}{y}$ at $(1,1)$

I'm trying to prove that the function $$f(x,y)=2xy+\frac{x}{y} $$ is differentiable at $(1,1)$. So I got: $$\nabla f(1,1)=\begin{bmatrix}3\\ 1 \end{bmatrix}$$ and: $$\lim_{(x,y)\rightarrow(1,1)...
1
vote
2answers
30 views

Computing partial derivatives using three implicitly defined equations

The three equations $x^2-y\operatorname{cos}(uv)+z^2=0$ $x^2+y^2-\operatorname{sin}(uv)+2z^2=0$ $xy-\operatorname{sin}u\operatorname{cosv}+z=0$ define $x,y,z$ as functions of $u,v$. Compute the ...
2
votes
2answers
394 views

Find a unit tangent vector to a curve that is an intersection of two surfaces.

The intersection of the two surfaces given by the Cartesian equations $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ passing through the point $P=(\sqrt{7},3,4)$. These equations may be ...
0
votes
1answer
35 views

Showing that $\nabla (\alpha f) = \alpha \nabla f$ for constant $\alpha$

I want show that del of alpha times a vector function for is equal to alpha times del of fun using. Alphar is a constant hence it should be factories out after finding partial derivetives,but how do ...
0
votes
1answer
124 views

If the integral of a vector field over a closed curve equals zero, is the field conservative?

If a vector field has a potential, then the integral of that vector field over every closed curve is zero. If the integral of a vector field over a closed curve equals zero, does that imply that the ...
0
votes
1answer
31 views

Differential of a vector field

How is the derivative of a vector field defined? Gradient only works on scalar fields, divergence or rotation is not what I am looking for. Let's take an easy example: $f: \mathbb R^2 \to \mathbb R^...
3
votes
2answers
35 views

relation between $\frac{\partial(x,0)}{\partial x}$ and $\left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$

if $u(x,t)$ differentiable function and i only have $u(x,0)$, then is it right $\frac{\partial(x,0)}{\partial x} = \left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$ or can i derive $u(x,0)$ to $x$ ...
0
votes
1answer
199 views

How to determine the maximum rate of increase in temperature

Suppose that the temperature at a point $(x,y,z)$ in space is given by $T(x,y,z)=\frac{80}{1+x^2+2y^2+3z^2}$ where $T$ is measured in degrees celsius and $x$,$y$ and $z$ in meters. In which ...
1
vote
1answer
24 views

Null velocity and piecewise smooth path

On texts of multivariable calculus and real analysis I have always seen the work made by $\mathbf{F}$ along the path $\gamma$ defined as the integral$$\int_\gamma\mathbf{F}\cdot\text{d}s:=\int_a^b\...
0
votes
1answer
46 views

What is the geometrical interpretation of these three integrals?

What is the geometrical interpretation of these three integrals? 1) $$\int_C~ (x+y) ~dx + (x^2y)~ dy$$ 2) $$\int_C~ \vec{F}\cdot d \vec{r}$$ 3) $$\int_C~ xyz~ ds$$ I know they are all line ...
2
votes
1answer
48 views

Change of variables in a double integral-proving injectivity

I want to make the following change: $$u=x^2 - y^2 \quad v=xy$$ where my region is in the first quadrant bounded by: $x^2-y^2 = 3 , \quad xy=1 , \quad x^2-xy-y^2 = 1 $ . How can I prove this change ...
2
votes
1answer
26 views

Showing $∇f (0, 0) = (0, 0)$ using chain rule

I'm trying to show the following but I'm not very sure how to proceed. Could someone please explain to me how to approach and solve the following question? Let $f : \Bbb R^2 → \Bbb R$ be ...
2
votes
1answer
61 views

Flux integral using Cartesian coordinates

Problem - need help for part (ii) Let $\vec{F} = y \vec{i} -x \vec{j} + z \vec{k}$ and let the surface $S$ be the part of the paraboloid $ z = 4 - x^2 - y^2$ with $z \geq 0 $, oriented with $\vec{n}$ ...
0
votes
1answer
23 views

Vector-valued function to describe a hyberboloid

I need to find a vector-valued function to describe the quadric surface $x^2+y^2-z^2=1$. I could use the identity $\cosh^2 u - \sinh^2 u = 1$, but I'm not sure how. The best I could arrive at is $\vec{...
2
votes
1answer
97 views

Integration of $|y|^{-2}$ over the ball $B(0,r)$

Can one explain why taking an integral of $\frac 1{|y|}$ over a ball in $\mathbb{R}^3$ of radius $r$ is equal to a constant times $r^2$? If $y \in \mathbb{R}^3$, then $$\int_{B(0,r)} \frac{dy}{|y|...
1
vote
1answer
38 views

What is the difference between these two integrals?

$1)$ $$\int_R 3 ~dxdy$$ $2)$ $$\iint_R 3xy~ dxdy$$ They both seem to be computed in the same way so I guess I don't understand why we use sometimes one integral and two other times.
-1
votes
1answer
66 views

What is the value of this triple integral?

I'm working through a question and it says: $$\int_{0}^{5}\int_{-z}^{z}\int_{0}^{\sqrt{z^2-y^2}}3xz~dxdydz=2500$$ Yet I cannot seem to get that answer so I'm wondering if this is a mistake?
0
votes
0answers
33 views

trying to prove that directional derivative is linear

let $f:\mathbb{R}^{2}\rightarrow\mathbb{R} $ differentiable at $x_0\in \mathbb{R}^2$ and let: $ d(v)=\begin{cases} D_{v}f(x_{0}) & v\neq0\\ 0 & v=0 \end{cases} $ I'm trying to prove ...
4
votes
2answers
19 views

Problems on orthogonality and tangency in 3-space.

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally. A cylinder whose equation is $y=f(x)$ is tangent to ...
2
votes
1answer
26 views

Let $\mathbf{r}=(x,y,z)$,$r=||\mathbf{r}||$. Show the following equation on $B\cdot \nabla (A\cdot \nabla (\frac{1}{r}))$

Let $\mathbf{r}=(x,y,z)$ and let $r=||\mathbf{r}||$. If $A$ and $B$ are constant vectors show that: $$B\cdot \left(\nabla \left (A\cdot \nabla \left(\frac{1}{r}\right)\right)\right)=\frac{3A\cdot \...
1
vote
3answers
61 views

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$.

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$ and $A$ and $B$ are constant vectors, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$. I'm a bit lost on this ...
0
votes
1answer
21 views

If $||\nabla f(x,y)||^2=2$, determine constants $a$ and $b$ such that $a(\frac{\partial g}{\partial u})^2-b(\frac{\partial g}{\partial v})^2=u^2+v^2.$

The change of variables $x=uv$, $y=\frac{1}{2}\left(u^2-v^2\right)$ transforms $f(x,y)$ to $g(u,v).$ If $\left\|\nabla f(x,y)\right\|^2=2$ for all $x$ and $y$, determine constants $a$ and $b$ such ...
2
votes
1answer
32 views

Concerning an application of the divergence theorem

I was studying the derivation of Helmholtz decomposition through Wikipedia and I've come across an application of the divergence theorem which I'm not familiar with. I'd appreaciate if you could help ...
1
vote
0answers
27 views

Why is this answer wrong. Double integral?

Let $R$ be the region bounded by the $4$ lines:$y=x+1, y+x-1, y=-x+1, y=-x-1$ Calculate $$\iint_R 3x^2 ~dxdy$$ My answer: $R=\{(x,y):-1 \leq x \leq 1, -1 \leq y \leq 1\}$ so then $$\iint_R 3x^2 ~...
0
votes
2answers
1k views

Changing variable in a second derivative

I want to convert the differentiation variable in a second derivative, but it's a bit more complicated than the case of the first derivative. For context, the variable ETA is a dimensionless density ...
1
vote
1answer
34 views

Gradient of a scalar function: path taken by a particle

Just the last part, I have no idea where to start.