MAT2131 - Coordinate Geometry and Vector Analysis¶

Coordinate and Polar System¶

Term 161¶

Q1(a) [4]: Find the equation of the plane through the points \((1,-2,2)\) and \((-3,1,2)\) and perpendicular to plane \(2x+y-z+6=0\).
Q1(b) [5]: Remove first degree terms from the equation \(3x^2+4y^2-12x+4y+13=0\).
Q1(c) [5]: Find the condition that the general equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) may represents a pair of straight lines.

Term 171¶

Q1(a) [3]: Find the relation between Cartesian coordinates and Polar coordinates.
Q1(b) [3]: Show that the area of the triangle formed by the straight lines \(y-2x=0\), \(y-3x=0\) and \(y=5x+4\) is \(\frac{4}{3}\).
Q1(c) [4]: Prove that a homogeneous quadratic equation \(ax^2+2hxy+by^2=0\) always represents a pair of straight lines which passing through the origin.
Q1(d) [4]: Show that the lines joining the origin to the points of intersection of \(7x^2+8xy-7y^2+6x-12y=0\) and \(2x+y-1=0\) are at right angles.
Q2(a) [5]: Prove that the straight lines represented by the equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) will be equidistance from the origin, if \(f^4-g^4=c(bf^2-ag^2)\).
Q2(b) [3]: Prove that the equation \(x^2+6xy+9y^2+4x+12y-5=0\) represent a pair of parallel lines.
Q3(a) [3]: Find the value of \(\lambda\) so that the equation \(\lambda x^2+4xy+y^2-4x-2y-3=0\) may represent a pair of straight lines.
Q3(b) [3]: If the pair of straight lines \(x^2-2mxy-y^2=0\) and \(x^2-2nxy-y^2=0\) be such that each pair bisects the angle between the other pair, prove that \(mn+1=0\).

Term 181¶

Q1(a): Find the condition that the general equation of the second degree \(ax^2+2hxy+by^2+2gx+2fy+c=0\) may represents a pair of straight lines.
Q1(b): By transforming to parallel axes through a properly chosen point \((h,k)\), prove that the equation \(12x^2-10xy+2y^2+11x-5y+2=0\) can be reduced to one containing only the terms of the second degree.
Q1(c): Show that the equation \(bx^2-2hxy+ay^2=0\) represents a pair of straight lines which are at right angle to the pair given by equation \(ax^2+2hxy+by^2=0\).
Q2(a): Find the value of k so that the equation \(6x^2+2kxy+12y^2+22x+31y+20=0\) may represent a pair of straight lines.
Q3(a): Test the nature of the conic given by the equation \(3x^2-8xy-3y^2+10x-13y+8=0\).
Q3(b): Find the equation of the sphere through the points \((0,0,0)\), \((1,-1,0)\), \((2,0,-2)\) and \((0,1,2)\).

Term 191¶

Q1(a) [3]: Find the relation between Cartesian coordinate and polar coordinate.
Q1(b) [5]: Origin shifted at \((4,-6)\). Find transformation of equation \(3x^2+5y^2+10xy+10=0\).
Q1(c) [6]: If the direction of axes is turned through an angle \(45^\circ\) and the origin remains unchanged then find the transformation equation of \(x^2-2xy+y^2+2x-4y+3=0\).

Term 201¶

Q1(a) [4]: Find the relation between Cartesian coordinates and Polar coordinates.
Q1(b) [4]: If the Axes are turned through \(45^\circ\), find the transformed form of the equation \(3x^2+3y^2+2xy=2\).
Q1(c) [6]: Show that the necessary condition of bisectors of the angles between the lines represented by \(ax^2+2hxy+by^2=0\) is \(\frac{x^2-y^2}{a-b}=\frac{xy}{h}\).
Q2(a) [3]: If the pair of straight lines \(x^2-2axy-y^2=0\) and \(x^2-2bxy-y^2=0\) be such that each pair bisects the angle between the other pair, prove that \(ab=-1\).
Q2(b) [5]: Prove that the straight lines represented by the equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) will be equidistance from the origin, if \(f^4-g^4=c(bf^2-ag^2)\).
Q3(a) [6]: Define circle. Show that the four points \((1,0)\), \((2,-7)\), \((8,1)\) and \((9,-6)\) are concyclic.
Q3(b) [3]: If \(ax^2+2hxy+by^2+2gx+2fy+c=0\) represents the non-degenerate conic, then write down the natures of conic.
Q3(c) [5]: For what value of \(\lambda\) the equation \(2xy+4x-6y+\lambda=0\) represent two intersecting lines? If \(\lambda=17\), then what type of conic represents by the given equation?

Term 211¶

Q1(a) [6]: Find the relation between Cartesian coordinates and polar coordinates. Transform to polar coordinates the equation \(x^3=y^2(2a-x)\).
Q1(b) [5]: Determine the equation of the parabola \(x^2-2xy+y^2+2x-4y+3=0\) after rotating of axes through \(45^\circ\).
Q1(c) [3]: Prove that a homogeneous quadratic equation \(ax^2+2hxy+by^2=0\) always represents a pair of straight lines which passing through the origin.
Q2(a) [3]: Show that the necessary condition of bisector of the angles between the lines represented by \(ax^2+2hxy+by^2=0\) is \(\frac{x^2-y^2}{a-b}=\frac{xy}{h}\).
Q2(b) [3+5]: Prove that the equation \(x^2+6xy+9y^2+4x+12y-5=0\) represent a pair of parallel lines.
Q2(c) [5]: Test the nature and find the center of the conic represented by \(2x^2-3xy+y^2-5x-4y+6=0\).
Q3(a): Find the area of the triangle formed by the lines \(ax^2+2hxy+by^2=0\) and \(lx+my+n=0\).
Q3(b): Find the equation of the sphere with its center at \((-4,2,3)\) and tangent to the plane \(2x-y+2z+7=0\).
Q3(c): Find the distance of the points \((2,0,1)\) and \((3,-3,2)\) from the plane \(x-2y+z=6\) and find whether the two points lie on the same side or opposite sides of the plane.

Polar Equation of Conics (Two and Three Dimension)¶

Term 171¶

Q2(c) [6]: Find the equation of the plane which is perpendicular to the plane \(5x+3y+6z+8=0\) and which contains the line of intersection of planes \(x+2y+3z-4=0\) and \(2x+y-z+5=0\).
Q3(c) [4]: Find the shortest distance between the straight lines through the points \(P(6,2,2)\) and \(Q(-4,0,-1)\) in the directions \((4,-2,2)\) and \((3,-2,-2)\) respectively.
Q3(d) [4]: Find the equation of the plane which is parallel to the plane \(2x-3y-6z-14=0\) and distance 5 units from the origin.

Term 191¶

Q2(a) [4]: Derive the equation of a circle in terms of polar coordinate.
Q2(b) [6]: Find the value of k for which the circles \(x^2+y^2-3x+ky-2=0\) and \(x^2+y^2+4x-5y-24=0\) are orthogonal.
Q2(c) [4]: Show that the polar equation of a conic can be written in the form \(r=\frac{l}{1-e\cos\theta}\).
Q4(a) [3]: Obtain the conditions of coplanarity of two straight lines in vector form.
Q5(a) [3]: Define direction cosines and direction ratios of a line.
Q5(b) [6]: Find the magnitude and the equation of shortest distance between the lines \(\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-4}{4}\) and \(\frac{x-2}{2}=\frac{y-4}{4}=\frac{z-5}{5}\).
Q5(c) [5]: Determine the angle between the line \(\frac{x-3}{6}=\frac{y-2}{3}=\frac{z+1}{-2}\) and the plane \(2x+y+2z+5=0\).

Term 201¶

Q2(c) [6]: Find the length of the shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\). State whether the lines are coplanar or not.

Vector Algebra and Its Applications to Geometry¶

Term 161¶

Q2(a) [8]: Find the length and equation of shortest distance (SD) between the lines \(\frac{x-1}{4}=\frac{y-2}{3}=\frac{z-1}{-5}\) and \(\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-4}{-4}\).
Q2(b) [6]: Obtain the equation of the plane which is perpendicular to the plane \(x+2y-3z+1=0\) and which contains the line of intersection of the planes \(3x+y+z-5=0\) and \(2x-3y+z+11=0\).
Q3(a) [3]: Prove that the magnitude A of the vector \(\vec A=A_1\hat i+A_2\hat j+A_3\hat k\) is \(A=\sqrt{A_1^2+A_2^2+A_3^2}\).
Q3(b) [3]: Determine the value of a, so that \(\vec A=2\hat i+a\hat j+\hat k\) and \(\vec B=4\hat i-2\hat j-2\hat k\) are perpendicular.
Q3(c) [4]: If \(\vec A\times\vec B=0\) and if A and B are not zero, show that A is parallel to B.
Q3(d) [4]: Prove that \(\vec A\cdot(\vec A\times\vec C)=0\).
Q4(a) [5]: Prove that \(\vec a\times(\vec b\times\vec c)+\vec b\times(\vec c\times\vec a)+\vec c\times(\vec a\times\vec b)=0\).
Q4(b) [3]: Find the vector equation of the line passing through the points \((3,1,1)\) and \((2,2,-1)\).

Term 171¶

Q4(a) [1+3]: What is position vector? The position vectors of points P and Q are given by \(\vec r_1=2\hat i+3\hat j-\hat k\) and \(\vec r_2=4\hat i-3\hat j+2\hat k\). Determine \(\overrightarrow{PQ}\) in terms of \(\hat i,\hat j,\hat k\) and its magnitude.
Q4(b) [4]: Define dot and cross product of two vectors. If \(\vec A=A_1\hat i+A_2\hat j+A_3\hat k\) and \(\vec B=B_1\hat i+B_2\hat j+B_3\hat k\), then show that \(\vec A\cdot\vec B=A_1B_1+A_2B_2+A_3B_3\).
Q4(c) [3]: Find the sine of the angle between the vectors \(\hat i+2\hat j+3\hat k\) and \(3\hat i-4\hat j+2\hat k\).
Q4(d) [3]: Find the vector equation of the line passing through the points \((3,1,1)\) and \((2,2,-1)\).

Term 181¶

Q2(b) [3+5]: Find the length and the equation of shortest distance between the lines \(\frac{x-1}{4}=\frac{y-2}{3}=\frac{z-1}{-5}\) and \(\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-4}{-4}\).
Q2(c): Show that the lines \(\frac{x+3}{2}=\frac{y+15}{3}=\frac{z-7}{-3}\) and \(\frac{x+1}{4}=\frac{y+1}{5}=\frac{z+1}{-1}\) are coplanar.
Q3(c): If a, b, c are three vectors, prove that \(\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c\).
Q3(d): Find the vector equation of the plane through the point \((3,-2,1)\) and perpendicular to the vector \(4\hat i-4\hat j+7\hat k\).
Q4(a): Find the angles which the vector \(\vec P=3\hat i-6\hat j+2\hat k\) makes with the coordinate axes.

Term 191¶

Q4(b) [6]: Find the projection of \(\vec a\) along \(\vec b\) and \(\vec b\) along \(\vec a\) where \(\vec a=2\hat i+6\hat j-3\hat k\) and \(\vec b=\hat i+4\hat j+8\hat k\).
Q4(c) [5]: If \(\vec a=4\hat i-\hat j-3\hat k\), \(\vec b=2\hat i+5\hat j\), and \(\vec c=\hat i+5\hat j-2\hat k\), then verify the vector triple product formula.

Term 201¶

Q4(a) [3]: The position vectors of points P and Q are given by \(\vec r_1=2\hat i+3\hat j-\hat k\), \(\vec r_2=4\hat i-3\hat j+2\hat k\). Determine \(\overrightarrow{PQ}\) in terms of \(\hat i,\hat j,\hat k\) and finds its magnitude.
Q4(b) [3]: Find the projection of the vector \(4\hat i-3\hat j+\hat k\) on the line passing through the points \((2,3,-1)\), \((-2,-4,3)\).
Q4(c) [4]: Find the angle which the vector \(\vec A=3\hat i-6\hat j+2\hat k\) makes with the coordinate axes.
Q4(d) [4]: Prove that \(\vec A\cdot(\vec B\times\vec C)=\vec B\cdot(\vec C\times\vec A)=\vec C\cdot(\vec A\times\vec B)\).
Q5(a) [2]: What are the physical significance of dot and cross product of vectors?

Term 211¶

Q4(a) [4]: What is vector? If D, E, F are middle points of sides BC, CA, AB of triangle ABC, then prove that \(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0\).
Q4(b) [3]: If \(\overrightarrow{OA}=2\hat i+3\hat j-4\hat k\) and \(\overrightarrow{OB}=4\hat i-3\hat j+2\hat k\), then find \(|\overrightarrow{AB}|\).
Q4(c) [4]: A force given by \(\vec F=3\hat i+2\hat j-4\hat k\) is applied at the point \((1,-1,2)\), find the moment of F about the point \((2,-1,3)\).

Vector Differentiation¶

Term 161¶

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 \(\nabla\cdot(\phi\vec A)=(\nabla\phi)\cdot\vec A+\phi(\nabla\cdot\vec A)\).

Term 171¶

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.

Term 181¶

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 \((1,0,1)\).

Term 191¶

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.

Term 201¶

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 \((0,1,2)\).

Term 211¶

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.

Vector Integration¶

Term 161¶

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.

Term 171¶

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.

Term 181¶

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.

Term 191¶

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 \(x+y+z=1\).

Term 201¶

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 \(y=x^2\).

Term 211¶

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 \((3,1,4)\).