MAT2131 - Coordinate Geometry and Vector Analysis¶

Vector Differentiation¶

Q4(c) [6]: Find the directional derivative of \(g=x^2yz^2+4y^2z\) at \((-1,-2,-1)\) in the direction \(5\hat i-2\hat j-3\hat k\).
Q5(a) [6]: If \(\vec F=(2xy+z^3)\hat i+(x^2+2y)\hat j+(3xz^2-2)\hat k\), show that F is conservative and find a scalar \(\phi\) such that \(\vec F=\nabla\phi\).
Q5(b) [4]: If \(\vec A=x^3z\hat i-2xyz\hat j+2yz^3\hat k\), find curl A at \((1,-1,1)\). Test whether this vector is irrotational or not.
Q5(c) [4]: Prove that \(\nabla\cdot(\phi\vec A)=(\nabla\phi)\cdot\vec A+\phi(\nabla\cdot\vec A)\).

Q5(a) [4]: If \(\vec V=\vec\omega\times\vec r\), prove that \(\vec\omega=\frac12\operatorname{curl}\vec V\), where \(\vec\omega\) is a constant vector.
Q5(b) [5]: Show that \(\nabla r^n=nr^{n-2}\vec r\), where \(\vec r=x\hat i+y\hat j+z\hat k\).
Q5(c) [5]: Find the divergence and curl of the vector field \(\vec F=x^2z\hat i-2yz^3\hat j+xy^2z\hat k\) at the point \((1,2,-1)\).
Q6(a) [3]: Determine the constant a so that \(\vec V=(3x^2+y)\hat i+(2y-z)\hat j+(x+az)\hat k\) is solenoidal.

Q4(b): Define gradient, divergence and curl.
Q4(c): Find the directional derivative of \(\phi=4e^{2x-y+z}\) at \((1,1,-1)\) in a direction toward the point \((-3,5,6)\).
Q4(d): Show that \(\vec r=(6xy+z^3)\hat i-(3x^2-z)\hat j+(3xz^2-y)\hat k\) is irrotational.
Q5(a): Find the velocity and acceleration of a particle which moves along the curve \(x=2\sin3t\), \(y=2\cos3t\), \(z=8t\) at any time \(t>0\); find the magnitude of the velocity and acceleration.
Q5(b): Determine the divergence and curl of the vector field \(\vec V=(2xz^3+6y)\hat i+(6x-2yz)\hat j+(3x^2z^2-y^2)\hat k\) at point \((-1,-2,1)\).
Q5(c) [3+3]: If \(\vec u=x^2z\hat i+yz^2\hat j-xy\hat k\), \(\vec v=xy^2\hat i-yz\hat j-zx\hat k\) and \(\phi=x^2yz\), find (i) \(\vec u\cdot(\nabla\phi)\) and (ii) \((\vec u\times\nabla\phi)\).
Q6(a) [1+3]: What is vector differential operator? Prove that \(\nabla^2\left(\frac1r\right)=0\), where \(\vec r=x\hat i+y\hat j+z\hat k\).
Q6(b): If \(\vec A=2x^2y\hat i-2xz\hat j+2y^2z\hat k\), find curl curl A at the point \((1,0,1)\).

Q3(a) [4]: Find the directional derivative of \(\phi=x^2yz^2+4y^2z\) at \((-1,-2,-1)\) in the direction \(5\hat i-2\hat j-3\hat k\).
Q3(b) [5]: Check whether \(\vec F=(2xy+z^3)\hat i+x^2\hat j+3xz^2\hat k\) is irrotational or solenoidal.
Q3(c) [5]: Suppose \(\vec v=\vec w\times\vec r\), show that \(\vec w=\frac12\operatorname{curl}\vec v\), where \(\vec w\) is a constant vector.

Q5(b) [4]: Find the directional derivative of \(\phi=x^2yz+4xz^2\) at \((1,-2,-1)\) in the direction \(2\hat i-\hat j-2\hat k\).
Q5(c) [4]: If \(\vec A=\cos(xy)\hat i+(3xy-2xz^2)\hat j-(3x+2y)\hat k\), show that \(\frac{\partial^2}{\partial y\partial x}(\vec A)=\frac{\partial^2}{\partial x\partial y}(\vec A)\).
Q5(d): A particle moves along the curve \(x=2t^2\), \(y=t^2-4t\), \(z=3t-5\), where t is the time. Find the components of its velocity and acceleration at time \(t=1\) in the direction \(\hat i-3\hat j+2\hat k\).
Q6(a) [5]: What is curl of a vector? If \(\vec V=\vec\omega\times\vec r\), prove that \(\vec\omega=\frac12\operatorname{curl}\vec V\), where \(\vec\omega\) is a constant vector.
Q6(b) [5]: When a vector is called solenoidal? Show that \(\vec A=(2x^2+8xy^2z)\hat i+(3x^3y-3xy)\hat j-(4y^2z^2+2x^3z)\hat k\) is not solenoidal but \(\vec B=xyz^2\vec A\) is solenoidal.
Q6(c) [4]: Find the angle between the surfaces \(z=x^2+y^2\) and \(z=2xy+e^x\) at the point \((0,1,2)\).

Q4(d): A particle moves along the curve \(x=2t^2\), \(y=t^2-4t\), \(z=3t-5\), where t is time. Find the components of its velocity and acceleration at time \(t=1\) in the direction \(\hat i-3\hat j+2\hat k\).
Q5(a): Define divergence and curl of a vector.
Q5(b): Find the directional derivative of \(\phi=4e^{2x-y+z}\) at \((1,1,-1)\) in a direction toward the point \((-3,5,6)\).
Q5(c): What is differential operator? Prove that \(\nabla^2\left(\frac1r\right)=0\), where \(\vec r=x\hat i+y\hat j+z\hat k\).
Q6(a): When a vector field is called irrotational? Find constants a, b, c so that \(\vec V=(x+2y+az)\hat i+(bx-3y-z)\hat j+(4x+cy+2z)\hat k\) is irrotational.
Q6(b): Show that if \(\vec A=(2x^2+8xy^2z)\hat i+(3x^3y-3xy)\hat j-(4y^2z^2+2x^3z)\hat k\) is not solenoidal but \(\vec B=xyz^2\vec A\) is solenoidal.
Q6(c): If \(\vec v=\vec\omega\times\vec r\), prove \(\vec\omega=\frac12\operatorname{curl}\vec v\), where \(\vec\omega\) is a constant vector.
Q6(d): Define ordinary derivatives of vector. A particle moves along a curve whose parametric equations are \(x=e^{-t}\), \(y=2\cos3t\), \(z=2\sin3t\), where t is the time. Determine its velocity and acceleration at any time.