0
votes
1answer
24 views

Trigonometric Partial Derivative

I need to find $$\frac{\partial Z}{\partial U} \text{ and } \frac{\partial Z}{\partial V}$$ for a $z=f(x,y) = \cos(xy) + y\cos(x)$. After a bit of an internet search, I think I have found the ...
0
votes
2answers
48 views

double partial differentiation

I'm having troubles with solving problems with partial differentiations... and this one is double. I don't thing we've even learned this in class... Question: If $z=f(x,y)$, where $x=r\cos(\theta), ...
0
votes
3answers
66 views

How does $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ make sense?

In Multivariable Calculus, the professor said that in order to compute the angle $x$ between two vectors $v$ and $w$, we use the formula: $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot ...
0
votes
1answer
72 views

Parametric Equation of sine wave helically wrapped around a cylinder

I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D). The parametric equation below gets me close to what I'm looking for, but not quite since the ...
4
votes
4answers
90 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
1
vote
0answers
20 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
0
votes
0answers
46 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
1
vote
1answer
419 views

Find all stationary points of multivariable function

$$f(x,y) = \left(y^2 + y -16\right)\sin(x)$$ Find ALL stationary points of $f$ and classify each as local max, min or saddle point. My working so far is $f_x = \left(y^2 + y -16\right)\cos x$ ...
0
votes
1answer
20 views

Given $\int\int_D \arctan \frac{y}{x}dxdy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates?

Given $\int\int_D \arctan \frac{y}{x}dxdy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates. I stuck with finding $\theta$. I know that $r ...
0
votes
2answers
82 views

Trigonometry gymnastics

The teacher is as usual jumping a million miles between steps, I appreciate if someone can break down how this step is done: $$\frac{\partial r}{\partial s}\times\frac{\partial r}{\partial t}=\det ...
0
votes
0answers
137 views

Directional derivative - angle between the vector and coordinate axes

Doing an exercise a about directional derivatives, it was required to find the derivative of a given function $f(x,y,z)$ in the direction of the vector $ \vec{v}$ that forms with the coordinates axes ...
0
votes
0answers
16 views

Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
0
votes
0answers
109 views

Trying to find coordinates of another point using bearings

I am trying to help one of my siblings with a trig project and there is one part that I am having a little trouble with. In his project, he is trying to find the coordinates of a UFO given the ...
1
vote
1answer
345 views

Finding the local extrema of this trigonometric, multivariate function

QUESTION Find all extrema and their places for $$ f(x,y) = \mathtt{sin} x + \mathtt{cos} y + \mathtt{cos} (x-y)$$ for $ 0 \le x \le \frac{\pi}{2}$ and $ 0 \le y \le \frac{\pi}{2}$ ATTEMPT I go ...
2
votes
1answer
96 views

Dot Product/ Cross Product Proof

Let $\hat{a}$, $\hat{b}$, and $\hat{c}$ $\in \mathbb{R}^3$, using the properties of vectors, prove $$ (\hat{a} \times \hat{b}) \cdot [(\hat{b} \times \hat{c}) \times (\hat{c} \times \hat{a})] = ...
1
vote
0answers
74 views

3d Implicit Trigonometry help?

I'm trying to understand implicit 3D trigonometry, specifically with this equation: $$\sin(y)+\cos(z)=\cos(x)$$ Can someone please explain to me what is going on with this equation? I really can't ...
2
votes
2answers
103 views

Is $\phi : \mathbb R^2 \rightarrow \mathbb R$ a differentiable function?

We have the following example on the book "Matrix Differential Calculus with Applications in Statistics and Econometrics", 3rd edition, p. 100. Let $\phi : \mathbb R^2 \rightarrow \mathbb R$ be a ...
3
votes
1answer
4k views

Intersection between a rectangle and a circle?

I have a poor working knowledge of math. I would like to calculate collision detection between a 2D circle and a 2D rectangle for a simple game of Pong. I thought of splitting the 2D rectangle into 4 ...
2
votes
2answers
764 views

Solve equation system with trigonometric functions

I need to maximize the function $$f(x,\theta) =x\sin\theta(xcos\theta + w - 2x)$$ which defines the area enclosed by a folded plate that forms a canal, where $w$ is the length of the plate, $x$ is the ...
0
votes
0answers
183 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...
1
vote
1answer
243 views

Taylor series expansion of $\sec(x +y^2)$

We have $f(x,y) = \sec(x+y^2)$ I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $ Using this ...
3
votes
1answer
574 views

Trig substitution for a triple integral

This problem involves calculating the triple integral of the following fraction, first with respect to $p$: $$ \int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} ...
6
votes
0answers
588 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f = 2\pi+2\tan^{-1}(y,x)$ $y = ...
2
votes
1answer
362 views

3d axis rotation

I have a vector V= and several line segments Seg1, Seg2, Seg3, Seg4. I want to know how to rotate each of the line segments so that the X axis is parallel to my given vector. How can I do this? ...
2
votes
4answers
12k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
6
votes
1answer
489 views

A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle

The coordinate transformation (due to Beukers, Calabi and Kolk) $$x=\frac{\sin u}{\cos v}$$ $$y=\frac{\sin v}{\cos u}$$ transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the ...