MAT2131 - Coordinate Geometry and Vector Analysis¶

Vector Integration¶

Q6(a) [5]: Find the line integral \(I=\oint(x\,dy-y\,dx)\) round the square \((0,0)\), \((1,0)\), \((1,1)\) and \((0,1)\), \((0,0)\).
Q6(b) [7]: If \(\vec F=y\hat i+(x-2xz)\hat j-xy\hat k\), evaluate \(\int_C\vec F\cdot d\vec r\) from \((0,0,0)\) to \((2,4,8)\) along the path C: (i) \(x=t\), \(y=t\), \(z=0\); (ii) the straight line joining \((0,0,0)\) to \((2,2,2)\).
Q6(c) [2]: State Green's theorem in the plane.
Q7(a) [4]: Apply Green's theorem to evaluate \(\oint_C[(2x^2-y^2)dx+(x^2+y^2)dy]\), where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle \(x^2+y^2=a^2\).
Q7(b) [4]: Evaluate \(\oint_C(3x+4y)dx+(2x-3y)dy\) where C, a circle of radius two with center at the origin of the xy plane, is traversed in the positive sense.
Q7(c) [2]: Give the representation of a point \(F(x,y,z)\) in cylindrical and spherical coordinates.
Q7(d) [4]: Prove that a cylindrical coordinate system is orthogonal.

Q6(c) [7]: If \(\phi=2xyz^2\), \(\vec F=xy\hat i-2\hat j+x^2\hat k\) and C is the curve \(x=t^2\), \(y=2t\), \(z=t^3\) from \(t=0\) to \(t=1\), evaluate the line integrals (i) \(\int_C\phi\,d\vec r\); (ii) \(\int_C\vec F\times d\vec r\).
Q7(a) [6]: State and prove Green's theorem in the plane.
Q7(b) [5]: Apply Green's theorem to evaluate \(\oint_C\{(3x-8y^2)dx+(4y-6xy)dy\}\), where C is the boundary enclosed by \(x=0\), \(y=0\) and \(x+y=1\).
Q7(c) [3]: Explain a curvilinear coordinate system.

Q6(c): Show that the line integral \(\int[(x^2-yz)dx+(y^2-zx)dy+(z^2-xy)dz]\) is independent of the path of integration between the points \((1,1,1)\) and \((2,3,4)\) and find its value.
Q7(a): State Green's theorem, Stokes theorem and the Gauss divergence theorem.
Q7(b): Evaluate the volume integral \(\iiint\operatorname{div}\vec F\,dv\) for the vector field \(\vec F=(2xy+z)\hat i+y^3\hat j-(e^x+3y)\hat k\) taken over the region bounded by \(2x+2y+z=6\), \(x=0\), \(y=0\), \(z=0\).
Q7(c): If \(\vec F=(xy-6x^2)\hat i+(2x-4x)\hat j\), evaluate \(\int_C\vec F\cdot d\vec r\) where C is the curve in the xy plane \(y=x^3\), from \((1,1)\) to \((2,8)\).
Q7(d): Write down curvilinear coordinates in polar, cylindrical and spherical coordinates.

Q6(a): If \(\psi=2xyz^2\), \(\vec B=xy\hat i-z\hat j+x^2\hat k\) and C is the curve \(x=t^2\), \(y=2t\), \(z=t^3\) from \(t=0\) to \(t=1\), evaluate the line integrals (i) \(\int_C\psi\,d\vec r\); (ii) \(\int_C\vec B\times d\vec r\).
Q6(b) [4]: Find the line integral \(I=\int(xdy-ydx)\) from \((0,0)\) to \((2,2)\) over the (i) straight line \(y=x\); (ii) square \((0,0)\) to \((1,0)\) to \((1,1)\) to \((0,0)\).
Q6(c) [5]: If \(\phi(x,y,z)=x^2y^3z\) and \(\vec A=xz\hat i-x^2y\hat j+y^2z^2\hat k\), then determine the magnitude of \(\frac{\partial^3}{\partial x^2\partial y}(\phi\vec A)\) at the point \((1,-1,2)\).
Q7(a) [3]: Give the idea of line, surface and volume integral.
Q7(b) [6]: Compute \(\oint_C x^2y\,dx+y^3\,dy\) by Green's theorem where C is the closed path formed by \(y=x\) and \(y^3=x^2\) from \((0,0)\) to \((1,1)\).
Q7(c) [5]: Evaluate \(\iiint\nabla\cdot\vec F\,dv\) where \(\vec F=xy\hat i+z^2\hat j+2xz\hat k\) bounded by the region \(x=0\), \(y=0\), \(z=0\) and the plane \(x+y+z=1\).

Q7(a): State and prove Green's theorem in the plane.
Q7(b) [4]: If \(\phi=2xyz^2\), \(\vec F=xy\hat i-z\hat j+x^2\hat k\) and C is the curve \(x=t^2\), \(y=2t\), \(z=t^3\) from \(t=0\) to \(t=1\), evaluate the line integrals (i) \(\int_C\phi\,d\vec r\); (ii) \(\int_C\vec F\times d\vec r\).
Q7(c): Verify Green's theorem in the plane for \(\oint_C(3x^2-8y^2)dx+(4y-6xy)dy\), where C is the boundary of the region defined by \(y=\sqrt{x}\) and \(y=x^2\).

Q5(d): If \(\vec A=(3x^2+6y)\hat i-14yz\hat j+20xz^2\hat k\), evaluate \(\int_C\vec A\cdot d\vec r\) from \((0,0,0)\) to \((1,1,1)\) along the following paths C: (i) \(x=t\), \(y=t^2\), \(z=t^3\); (ii) the straight lines from \((0,0,0)\) to \((1,0,0)\) then \((1,1,0)\).
Q7(a): State and prove Green's theorem.
Q7(b): Verify Green's theorem in the plane for \(\oint_C(xy+y^2)dx+x^2dy\), where C is the closed curve of the region bounded by \(y=x\) and \(y=x^2\).
Q7(c): Show that \(\vec F=(2xy+z^3)\hat i+x^2\hat j+3xz^2\hat k\) is a conservative force field; also find the total work done in moving an object in this field from \((1,-2,1)\) to \((3,1,4)\).