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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7 views

Tangent Vector at the point $(1,2,11)$ whose projection onto the $xy$-plane is parallel to vector $1/\sqrt{10}i+3/\sqrt{10}j$.

$$f(x,y) = x^3y^2 + 3x + 2y$$ The gradient of $f$ at the point $(1,2)$ is $15{\bf i} + 6{\bf j}$. ${\bf u} = \dfrac{1}{\sqrt{10}}{\bf i} + \dfrac{3}{\sqrt{10}}{\bf j}$ The Directional Derivative ...
4
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1answer
38 views

Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $\nabla^2 T = \partial ...
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1answer
23 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
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1answer
21 views

Construct a procedure which determines the location of the shadow of a rectangluar box.

I drew a 3d rectangular box on a coordinate plan consisting of x, y, and z. A procedure is to be created that will determine the location of the shadow of the box on one of the coordinate planes. I ...
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2answers
65 views
+50

Integration of the vector field $\mathbf {F } (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j $ over two ellipses

Let $\mathbf{F}$ be a vector field defined on $\mathbb R^2 \setminus\{(0,0)\}$ by $$\mathbf {F } (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j $$ Let $\gamma,\alpha:[0,1]\to\mathbb R^2$ be defined by ...
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0answers
26 views

How to calculate the volume of a tetrahedron?

Suppose that $$ I=\iiint_{V}f(x,y,z)dxdydz $$ where $f(x,y,z)$ is a continuous function, $V$ is a tetrahedron whose vertices are $P(2,2,0), A(-2,0,0), B(0,0,2)$ and $C(1,1,3)$. I want to ask you how ...
0
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1answer
27 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
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0answers
19 views

Geometric position of gradient on the surface in $\mathbb{R}^3$ and orthogonality to tangent of level curve

Given a function $f(x, y)\in C^1(\mathbb{R}^2)$ and its gradient $\nabla f(x, y) =(\frac{\partial f(x, y)}{\partial x}, \frac{\partial f(x, y)}{\partial y})$ which forms a vector field where each ...
2
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1answer
25 views

Calculate surface integral

I need some help with the following: Given $$f(x,y,z)=\left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac32}}, \frac{-z}{(x^2+y^2+z^2)^{\frac32}} \right),$$ calculate the ...
0
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1answer
111 views

What is the effect of axis rotation on functions defined on $\mathbb{R}^{2}$

I haven't studied multivariable calculus yet but I have a question that bothers me. Let $F$ be a function $\mathbb{R}^2 \to \mathbb{R}$. Imagine that we rotate the co-ordinate axis by an angle ...
4
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1answer
28 views

Independence of function and its derivative in calculus of variations

It's common to see in calculus of variation that the integrand $f$ of functional $F[y]=\int f(y,y',x)dx$ is a function of $y,y'$ and $x$. Why do we regard the derivative $y'$ as an independent ...
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1answer
36 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
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0answers
37 views

To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
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1answer
20 views

How can you find the distance between two skew planes?

I understand that there is a unique line perpendicular to both planes and the length of that line is the distance between the planes but how would I go about finding the what the equation of that line ...
4
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1answer
183 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
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1answer
26 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
1
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1answer
341 views

Why gradient vector is perpendicular to the plane

I know what gradient vector or $∇F$ is and I know how to prove it is orthogonal to the surface (using calculation - not intuitive). In a particular case, in which we have a three variable function, I ...
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3answers
36 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
3
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1answer
19 views

Chain rule notation for composite functions

Suppose I have a function $ f(x, y, g(x, y)) $ How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + ...
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2answers
33 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
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0answers
23 views

Finding the normal vector of a surface (Flux of a vector field n*dS expression)

This problem is practice for a final exam. Let S be the closed surface whose bottom face B is the unit disc in the $xy$-plane and whose upper surface U is the paraboloid $ z = 1 − x^2 − y^2 , z \geq ...
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1answer
41 views

Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex). Any hint?
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3answers
226 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
1
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1answer
43 views

Evaluate $\int_0^1\int_x^1 e^{x/y} dy\,dx$

I need some help to solve the following: $$\int_0^1\int_x^1 e^{x/y} dy\,dx$$ I guess it is related with change of variable, but I can't figure out which one. Thanks in advance. Regards.
0
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1answer
24 views

Minimize squared distance to origin from a paraboloid

I have to use Lagrange multilpiers to find the minimum distance from the paraboloid with equation $z = \left({x-1/}{\sqrt{2}}\right)^2 + \left({y-1/}{\sqrt{2}}\right)^2$ to the origin, and from this ...
1
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2answers
84 views

Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$

Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact ...
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0answers
28 views

Second derivative of the position vector in a spherical coordinate system

In a spherical coordinate system my unit vectors are: $\vec{e_r}=\begin{pmatrix}\sin\theta\cdot \cos\phi \\ \sin\theta \cdot \sin\phi \\ \cos\theta \end{pmatrix}$; ...
1
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1answer
38 views

Notation of multivariable derivatives

I reading a text where quadratic terms of a function $L : \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^2$ are expanded as $$ L(x,\alpha)=\frac{1}{2}L_{20,0} \, x_0^2 + L_{20,1} \, x_0 x_1 + ...
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2answers
22 views

How to evaluate the gradient of a function at a point

I have a problem where I am to create a function in terms of $x$ and $y$ and compute the gradient at the point $(1,1)$. I computed the gradient but in order to evaluate it at the given point do I just ...
2
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1answer
85 views

Prove equality of two vectors if they have equal divergence and equal curls

I have following question: Fields with equal divergence and equal curls $F_1$ and $F_2$ are two vectors fields, you may write them as $F_1 = M_1i+N_1j+P_1k$, $F_2 = M_2i+N_2j+P_2k$. Suppose that ...
2
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2answers
266 views

Finding extreme values when the determinant of the Hessian at a critical point is zero.

We want to determine extreme values of $f(x,y)=x^3+xy^2-x^2y-y^3$. We first determine critical points by solving $\dfrac{\partial f(x,y)}{\partial x}=0$ and $\dfrac{\partial f(x,y)}{\partial y}=0$ ...
2
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2answers
53 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
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0answers
22 views

Theorem proving skills in calculus, clearer idea to read in reverse order; linear-reading with writing down helps little

It is said that theorem proving skills are better trained via reproducing proofs from sketch rather than passive reading. Here we need more precise extension. e.g. in Multivariable Calculus, there ...
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0answers
32 views

Create a fourth order polynomial function f(x,y) with at least two distinct terms

I will be computing the gradient, finding the critical points, and use Lagrange multipliers to either maximize or minimize the function. Any suggestions?
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1answer
26 views

Verify that every tangent plane to the cone $x^2+y^2-z^2 = 0$ passes through the origin.

I'm supposed to show that every tangent plane to the cone $x^2+y^2-z^2 = 0$ passes through the origin. I set $f(x) = x^2 + y^2 - z^2$ and calculated the gradient of f. $\nabla f = \langle ...
0
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1answer
26 views

Two ways of finding a Potential of a Vector Field

If $\vec{F}(x,y)$ is a conservative vector field and we want to find a function $V$ such that $\nabla(V)=\vec{F}$, then one way to do it is to take an arbitrary point $(x_0,y_0)$ and then define ...
4
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0answers
48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
2
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1answer
35 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
2
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1answer
50 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
5
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2answers
28 views

conical surface, parametrization, immersion, Gaussian and mean curvatures

"Find the parametric form of a conical surface $S$ which is spanned by all rays starting $($but not including $)$ a fixed point $\gamma$ and passing through an arbitrary point on $\gamma$ and passing ...
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2answers
271 views

Find the local max/mins of $f(x, y) = xy + 8/x + 8/y$

I get no local maximums, a local minimum at $(2, 2)$, and no saddle points. Supposedly my answer for local minimum is incorrect and I keep making this same error. Please show me how to find the local ...
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0answers
44 views

Surface Area in 4 dimensions [on hold]

We all know the surface area relation in 3 space from multivariable calculus, however, how is this taken to 4 space?
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1answer
56 views

Composition of multiple variable limits

I need help to prove the following theorem: Let $f,g$ be functions, $a \in D(f \circ g)$ a limit point. If $\lim_{x \to a} f(x)=b$ and $\lim_{y \to b} g(y)=c$, then $\lim_{x \to a} g(f(x)) = c$, if ...
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4answers
144 views

Show that the series $\sum_{n,m=1}^\infty 1/(n+m)!$ is absolutely convergent and find its sum

Show that the series $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!}$$ is absolutely convergent and find its sum. This comes from a chapter called interchange of limit operations. I tried using the ratio ...
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vote
1answer
20 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
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2answers
392 views

A question on notation: What does $\nabla |\vec{a} \times \vec{r}|^n$ mean?

I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly. I am trying to evaluate the following, namely $\nabla w = ...
0
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1answer
23 views

What space curves can this theorem describe?

We were given the following theorem in our Vector Calculus class: THM: For space curve $R$ which does not pass through the origin, and which has a second derivative, the following are equivalent: ...
2
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0answers
38 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
3
votes
1answer
36 views

Using Stokes' Theorem to evaluate $\displaystyle\int_{C}{(xyz)dx+(xy)dy+(x)dz}$

Let $C$ be the closed, piecewise smooth curve formed by traveling in straight lines between the points $(0,0,0),(2,1,5),(1,1,3)$ and back to the origin, in that order. Use Stokes' theorem to evaluate ...
0
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2answers
35 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...