Change of coordinatesSecond derivative of the position vector in a spherical coordinate systemFollow up question: Differential equation math puzzleTriple integral in spherical coordinates: Finding $phi$ limits?Triple integral - converting to cylindrical coordinatesSimplifying Difference QuotientVolume of the solid bounded by the region $E = (x,y,z) , : , x^2 + y^2 +z^2 - 2z leq 0, sqrtx^2+y^2 leq z$Interchange $phi$ and $theta$ in spherical coordinatesDeriving laplacian in spherical coordinates using matrix operationsSpherical coordinates to calculate triple integralConverting cartesian rectangular equation to it's corresponding polar equation
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Change of coordinates
Second derivative of the position vector in a spherical coordinate systemFollow up question: Differential equation math puzzleTriple integral in spherical coordinates: Finding $phi$ limits?Triple integral - converting to cylindrical coordinatesSimplifying Difference QuotientVolume of the solid bounded by the region $E = (x,y,z) , : , x^2 + y^2 +z^2 - 2z leq 0, sqrtx^2+y^2 leq z$Interchange $phi$ and $theta$ in spherical coordinatesDeriving laplacian in spherical coordinates using matrix operationsSpherical coordinates to calculate triple integralConverting cartesian rectangular equation to it's corresponding polar equation
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Hi everyone!
Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...
The other parts were correct.
I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$
calculus
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add a comment |
$begingroup$
Hi everyone!
Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...
The other parts were correct.
I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$
calculus
$endgroup$
add a comment |
$begingroup$
Hi everyone!
Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...
The other parts were correct.
I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$
calculus
$endgroup$
Hi everyone!
Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...
The other parts were correct.
I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$
calculus
calculus
edited 2 days ago
Ingix
4,867159
4,867159
asked 2 days ago
aluno20000aluno20000
446
446
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.
That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.
That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:
1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.
2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.
Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.
So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!
$endgroup$
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.
That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.
That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:
1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.
2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.
Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.
So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!
$endgroup$
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
add a comment |
$begingroup$
You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.
That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.
That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:
1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.
2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.
Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.
So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!
$endgroup$
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
add a comment |
$begingroup$
You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.
That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.
That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:
1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.
2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.
Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.
So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!
$endgroup$
You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.
That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.
That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:
1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.
2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.
Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.
So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!
answered 2 days ago
IngixIngix
4,867159
4,867159
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
add a comment |
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
$begingroup$
Thank you for the answer!
$endgroup$
– aluno20000
2 days ago
add a comment |
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