Multivariable Calculus

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Vector Calculus

Vector Differentiation

Solution If A=xyz{\vec  i}+xz^{2}{\vec  j}-y^{3}{\vec  k},B=x^{3}{\vec  i}-xyz{\vec  j}+x^{2}z{\vec  k}\, calculate {\frac  {\partial ^{2}{\vec  A}}{\partial y^{2}}}\times {\frac  {\partial ^{2}{\vec  B}}{\partial x^{2}}} at the point (1,1,0).
Solution Find {\frac  {dr}{dt}},{\frac  {d^{2}r}{dt^{2}}} when r=3i-6t^{2}j+4tk

Solution If r=\sin ti+\cos tj+tk\, Find {\frac  {dr}{dt}},{\frac  {d^{2}r}{dt^{2}}},\left|{\frac  {dr}{dt}}\right\vert ,\left|{\frac  {d^{2}r}{dt^{2}}}\right\vert

Solution If r=\cos nti+\sin ntj\, where n is constant and t varies, prove that r\times ({\frac  {dr}{dt}})=nk\, and r\cdot ({\frac  {dr}{dt}})=0\,.

Solution If r=e^{{nt}}a+e^{{-nt}}b\, where a,b are constant vectors, show that ({\frac  {d^{2}r}{dt^{2}}})-n^{2}r=0\,

Solution If r=a\cos \omega t+b\sin \omega t\,,Show that r\times {\frac  {dr}{dt}}=\omega a\times b\, and {\frac  {d^{2}r}{dt^{2}}}=-\omega ^{2}r\,

Solution If u=t^{2}i-tj+(2t+1)k\, and v=(2t-3)i+j-tk\,, Find {\frac  {d}{dt}}(u\cdot v)\, and {\frac  {d}{dt}}(u\times v)\, where t=1.

Solution If a=\sin \theta i+\cos \theta j+\theta k,b=\cos \theta i-\sin \theta j-3k,c=2i+3j-k\,.Find {\frac  {d}{d\theta }}[a\times (b\times c)]\, at \theta =0\,

Solution A particle moves along a curve whose parametric equations are x=e^{{-t}},y=2\cos 3t,z=\sin 3t\,.Find the velocity and acceleration at t=0.

Solution A particle moves along the curve x=t^{3}+1,y=t^{2},z=2t+5\, where t is the time.Find the components of its velocity and acceleration at t=1 in the direction of i+j+3k\,.

Solution A particle moves so that its position vector is given by r=\cos \omega ti+\sin \omega tj\, where \omega \, is a constant.Show that i).The velocity of the particle is perpendicular to r ii).The acceleration is directed towards the origin and has magnitude proportional to the distance from the origin.

Solution Show that if a,b,c are constant vectors,then r=at^{2}+bt+c\, is the path of a particle moving with constant acceleration.

SolutionIf f=\cos xyi+(3xy-2x^{2})j-(3x+2y)k\,,find the value of {\frac  {\partial f}{\partial x}},{\frac  {\partial f}{\partial y}},{\frac  {\partial ^{2}f}{\partial x^{2}}},{\frac  {\partial ^{2}f}{\partial y^{2}}},{\frac  {\partial ^{2}f}{\partial x\partial y}}\,.

Solution If f=(2x^{2}y-x^{4})i+(e^{{xy}}-y\sin x)j+x^{2}\cos yk\,.Verify that {\frac  {\partial ^{2}f}{\partial x\partial y}}={\frac  {\partial ^{2}f}{\partial y\partial x}}\,.

Solution If \phi (xyz)=xy^{2}z\, and f=xzi-xyj+yz^{2}k\,.Find {\frac  {\partial ^{3}(\phi f)}{\partial x^{2}\partial z}}\, at (2,-1,1).

Vector Integration

Solution If f(t)=(t-t^{2})i+2t^{3}j-3k\,,Find i).\int f(t)\,dt\, ii).\int _{1}^{2}f(t)\,dt\,

Solution If f(t)=ti+(t^{2}-2t)j+(3t^{2}+3t^{3})k\,,find \int _{0}^{1}f(t)\,dt\,.

Solution Evaluate \int _{0}^{1}(e^{t}i+e^{{-2t}}j+tk)\,dt\,

Solution If r=ti-t^{2}j+(t-1)k\, and s=2t^{2}i+6tk\,,Evaluate \int _{0}^{2}r\cdot s\,dt\, and \int _{0}^{2}r\times s\,dt\,.

Solution Evaluate \int _{0}^{2}a\cdot b\times c\,dt\, where a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,

Solution Given that r(t)=2i-j+2k\, when t=2,r(t)=4i-2j+3k\, when t=3, Show that \int _{2}^{3}[r\cdot {\frac  {dr}{dt}}]\,dt=10\,.

Solution Evaluate \int _{1}^{2}r\times {\frac  {d^{2}r}{dt^{2}}}\,dt\, where r=2t^{2}i+tj-3t^{3}k\,

Solution If r(t)=5t^{2}i+tj-r^{3}k\,,prove that \int _{1}^{2}r\times {\frac  {d^{2}r}{dt^{2}}}\,dt=-14i+75j-15k\,

Solution Evaluate \int a\cdot [r\times {\frac  {d^{2}r}{dt^{2}}}]\,dt\,

Solution Evaluate \int _{1}^{2}[a\cdot (b\times c)+a\times (b\times c)]\,dt\, where a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,

Solution The acceleration of a moving particle at any time t is given by {\frac  {d^{2}r}{dt^{2}}}=12\cos 2ti-8\sin 2tj+16tk\,.Find the velocity v and displacement r at any time t,if t=0,v=0 and r=0.

Solution Find the value of r satisfying the equation {\frac  {d^{2}r}{dt^{2}}}=6ti-24t^{2}j+4\sin tk\, given that r=2i+j,{\frac  {dr}{dt}}=-i-3k\, at t=0.

Solution If the acceleration of a particle at any time t greater than or equal to zero is given by a=3\cos ti+4\sin tj+t^{2}k\, and the velocity v and displacement r are zero at t=0, then find v and r at any time t.

Solution Integrate {\frac  {d^{2}r}{dt^{2}}}=-n^{2}r\,

Gradient Divergence and Curl

SolutionIf f(x,y,z)=x^{3}+y^{3}+z^{3}+3xyz\, then find \nabla f\,

SolutionIf f(x,y,z)=3x^{2}y-y^{3}z^{2}\,,find \nabla f\, at the point (1,-2,-1).

Solution If r=xi+yj+zk\, and r=|r|=(x^{2}+y^{2}+z^{2})^{{{\frac  {1}{2}}}}\,, Prove that

i). \nabla f(r)=f'(r)\nabla f\, ii). \nabla r=({\frac  {1}{r}})r\,

Solution If \phi (x,y,z)=(3r^{2}-4r^{{{\frac  {1}{2}}}}+6r^{{-{\frac  {1}{3}}}})\,,Show that \nabla \phi =2(3-r^{{-{\frac  {3}{2}}}}-r^{{-{\frac  {7}{3}}}})r\,

Solution If u=x+y+z,v=x^{2}+y^{2}+z^{2},w=xy+yz+zx\, Prove that {\mathrm  {grad}}u\cdot [\nabla v\times \nabla w]=0\,

Solution Evaluate \nabla e^{{r^{2}}}\, where r^{2}=x^{2}+y^{2}+z^{2}\,

Solution Show that (a\cdot \nabla )\phi =a\cdot \nabla \phi \,

Solution If F=[y{\frac  {\partial f}{\partial z}}-z{\frac  {\partial f}{\partial y}}]i+[z{\frac  {\partial f}{\partial x}}-x{\frac  {\partial f}{\partial z}}]j+[x{\frac  {\partial f}{\partial y}}-y{\frac  {\partial f}{\partial x}}]k\,,Prove that i).F=r\times \nabla f\, ii).F\cdot r=0\, iii).F\cdot \nabla f=0\,

SolutionIf u=3x^{2}y,v=xz^{2}-2y\,,find (\nabla u)\cdot (\nabla v)\,

SolutionFind \nabla \phi ,|\nabla \phi |\, where \phi (x,y,z)=(x^{2}+y^{2}+z^{2})e^{{-(x^{2}+y^{2}+z^{2})^{{{\frac  {1}{2}}}}}}\,

SolutionIf f=x^{2}yi-2xzj+2yzk\,, find i). divf\,. ii). Evaluate div[(x^{2}-y^{2})i+2xyj+(y^{2}-2xy)k]\,

Solution If a_{1}i+a_{2}j+a_{3}k\,,prove that \nabla \cdot a=(\nabla a_{1})\cdot i+(\nabla a_{2})\cdot j+(\nabla a_{3})\cdot k\,

SolutionIf a=(x+3y)i+(y-3z)j+(x-2z)k\,,find (a\cdot \nabla )a\,

Solution Evaluate \nabla \cdot (a\times r)r^{n}\, where a is a constant vector.

Solution Find \nabla \times f\, or curl F, where

i). F=x^{2}yi-2xzj+2yzk\, ii). F=(x^{2}-y^{2})i+2xyj+(y^{2}-2xy)k\,

Solution Prove that curlcurlF=0\, where F=zi+xj+yk\,

Solution If V=e^{{xyz}}(i+j+k)\,,find curlV\,

Solution If r=xi+yj+zk\,,prove that i).divr=3\,

ii). If r=xi+yj+zk\, show that curlr=0\,

Solution IF f=xy^{2}i+2x^{2}yzj-3yz^{2}k\,,Find {\mathrm  {div}}f,{\mathrm  {curl}}f\,.What are their values at(1,-1,1)\,

Solution Find the {\mathrm  {curl}}\, of the vector V=(x^{2}+yz)i+(y^{2}+zx)j+(z^{2}+xy)k\, at the point(1,2,3)\,

Solution If f=(x+y+1)i+j+(-x-y)k\,,prove that f\cdot {\mathrm  {curl}}f=0\,

Solution a). Prove that vector f=(x+3y)i+(y-3z)j+(x-2z)k\, is solenoidal.

b). Determine the constant 'a' so that the vector f=(x+3y)i+(y-2z)j+(x+az)k\, is solenoidal.

Solution a). Show that the vector f=(\sin y+z)i+(x\cos y-z)j+(x-y)k\, is irrational. b). Determine the constants 'a','b','c' so that the vector f=(x+2y+az)i+(bx-3y-z)j+(4x+cy+2z)k\, is irrational.

Solution Prove that \nabla \cdot (r^{3}r)=6r^{3}\,

Solution Prove {\mathrm  {div}}[r\nabla r^{{-3}}]=3r^{{-4}}\, or \nabla \cdot [r\nabla ({\frac  {1}{r^{3}}})]={\frac  {3}{r^{4}}}\,

Solution If a is a constant vector,prove that {\mathrm  {curl}}{\frac  {a\times r}{r^{3}}}=-{\frac  {a}{r^{3}}}+{\frac  {3r}{r^{5}}}(a\cdot r)\,

Solution Show that \nabla ^{2}({\frac  {x}{r^{3}}})=0\,

Solution Show that {\mathrm  {div}}{\mathrm  {grad}}(r^{m})=m(m+1)r^{{m-2}}\,

Solution Evaluate {\mathrm  {curl}}{\mathrm  {grad}}(r^{m})\, where r=|r|=|xi+yj+zk|\,

Solution If u=x^{2}-y^{2}+4z\,,Show that \nabla ^{{2}}u=0\,

Solution Show that u=ax^{2}+by^{2}+cz^{2}\, satisfies Laplace equation \nabla ^{2}u=0\,

Solution If f and g are two scalar functions,prove that {\mathrm  {div}}(f\nabla g)=f\nabla ^{2}g+\nabla f\times \nabla g\,

Solution Show that \nabla \cdot (\nabla \times r)=0\, if \nabla \times V=0\,

Solution Prove that \nabla ^{2}({\frac  {1}{r}})=0\,,where r^{2}=x^{2}+y^{2}+z^{2}\,

Solution Evaluate \nabla ^{2}({\frac  {x}{r^{2}}})\,

Solution Prove that V\times {\mathrm  {curl}}V={\frac  {1}{2}}\nabla V^{2}-(V\cdot \nabla )V\,

SolutionIf v=v_{1}i+v_{2}j+v_{3}k\,,prove that \nabla \times v=\nabla v_{1}\times i+\nabla v_{2}\times j+\nabla v_{3}\times k\,

Solution If r(P) be the vector from the origin O to a point P in the xy-plane,then show that the plane scalar field u(P)=\log r\, satisfies the equation \nabla ^{2}u=0\,

Solution Prove that {\mathrm  {div}}(A\times r)=r\cdot {\mathrm  {curl}}A\,

Solution Prove that \nabla \times (F\times r)=2F-(\nabla \cdot F)r+(r\cdot \nabla )F\,

Solution If u=e^{{2x}}+x^{2}z\, and v=2z^{2}y-xy^{2}\,,find {\mathrm  {grad}}(uv)\, at the point (1,0,2).

Solution Prove that {\mathrm  {curl}}[r\times (a\times r)]=3r\times a\,,where a is a constant vector.

Solution Find the unit normal to the surface z=x^{2}+y^{2}\, at the point (-1,-2,5).

Solution Find the directional derivative of \phi =x^{2}yz+2xz^{2}\, at (1,-2,-1) in the direction of 2i-j-2k\,

Solution Calculate the maximum rate of change and the corresponding direction for the function \phi =x^{2}y^{3}z^{4}\, at the point 2i+3j-k\,

Solution Find the equation of the tangent plane and normal to the surface xyz=4\, at the point (1,2,2).

Solution Find the equation of the tangent line and normal plane to the curve of intersection of x^{2}+y^{2}+z^{2}=1,x+y+z=1\, at (1,0,0).

Solution Find the angle between the curves x^{2}+y^{2}+z^{2}=9,z=x^{2}+y^{2}-3\, at the point (2,-1,2).

Solution Find the constants a and b so that surfaces ax^{2}-byz=(a+2)x\, will be orthogonal to the surface 4x^{2}y+z^{3}=4\, at the point (1,-1,2).

Line, Surface & Volume Integrals

Solution If F=3xyi-y^{2}j\,,Evaluate \int _{{C}}F\cdot dr\,\,,where C is the curve y=2x^{2}\, in the xy-plane from (0,0)to (1,2).

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=(x^{2}+y^{2})i-2xyj\,,C is the rectangle in xy-plane bounded by y=0,x=a,y=b,x=0\,

Solution If F=(2x+y)i+(3y-x)j\,,evaluate \int _{{C}}F\cdot \,dr\, where C is the curve in the xy-plane consisting of the straight line from (0,0)\, to (2,0)\, and then to (3,2)\,

Solution If F=(3x^{2}+6y)i-14yzj+20xz^{2}k\,,evaluate \int _{{C}}F\cdot \,dr\, where C is the straight line joining (0,0,0),(1,1,1)\,

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=\cos yi-x\sin yj\, and C is the curve y={\sqrt  {1-x^{2}}}\, in the xy-plane from (1,0)\, to (0,1)\,

Solution Evaluate \int F\cdot \,dr\, along the curve x^{2}+y^{2}=1,z=1\, in the positive direction from (0,1,1) to (1,0,1),where F=(2x+yz)i+xzj+(xy+2z)k\,

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=x^{2}y^{2}i+yj\, and the curve c is y^{2}=4x\, in the xy-plane from (0,0) to (4,4) where r=xi+yj\,

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=xyi+(x^{2}+y^{2})j\, and C in the arc of the curve y=x^{2}-4\, from (2,0) to (4,12).

Solution If F=(2x^{2}+y^{2})i+(3y-4x)j\, evaluate \int F\cdot \,dr\, around the triangle ABC whose vertices are A(0,0),B(2,0),C(2,1)\,

Solution If F=(2y+3)i+xzj+(yz-x)k\,.evaluate \int _{{C}}F\cdot \,dr\, where C is the path consisting of the straight lines from (0,0,0) to (0,0,1) then to (0,1,1) and then to (2,1,1).

Solution If A=(2y+3)i+xzj+(yz-x)k\,,evaluate \int _{{C}}A\cdot \,dr\, along the curve C.x=2t^{2},y=t,z=t^{3}\, from t=0 to t=1.

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=zi+xj+yk\, and C is the arc of the curve r=\cos ti+\sin tj+tk\, from t=0\, to t=\pi \,

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=yzi+zxj+xyk\, and C is the portion of the curve r=a\cos ti+b\sin tj+ctk\, from t=0\, to t={\frac  {\pi }{2}}\,

Solution Evaluate \int _{{C}}F\cdot \,dr\, where F=xyi+yzj+zxk\, and C is the arc of the curve r=a\cos \theta i+a\sin \theta j+a\theta k\, from \theta =0\, to \theta ={\frac  {\pi }{2}}\,

Solution If F=xyi-zj+x^{2}k\,,evaluate \int _{{C}}F\times \,dr\, where C is the curve x=t^{2},y=2t,z=t^{3}\, from t=0 to 1.

Solution Find the total work done in moving a particle in a force field given by F=3xyi-5zj+10xk\, along the curve x=t^{2}+1,y=2t^{2},z=t^{3}\, from t=1 to t=2.

Solution Find the work done when a force F=(x^{2}-y^{2}+x)i-(2xy+y)j\, moves a particle in xy-plane from (0,0) to (1,1) along the curve y^{2}=x\,

Solution Find the work done in moving a particle in a force field F=3x^{2}i+(2xz-y)j+zk\, along the line joining (0,0,0) to (2,1,3).

Solution Find the work done in moving a particle once round a circle C in the xy-plane,if the circle has centre at the origin and radius 3 and when the force field is given by F=(2x-y+z)i+(x+y-z^{2})j+(3x-2y+4z)k\,

Solution Find the circulation of F round the curve C where F=yi+zj+zxk\, and C is the circle x^{2}+y^{2}=1,z=0\,

Solution Find the circulation of F round the curve C where F=e^{x}\sin yi+e^{x}\cos yj\, and C is the rectangle whose vertices are (0,0),(1,0),(1,{\frac  {\pi }{2}}),(0,{\frac  {\pi }{2}})\,

Solution Evaluate \iint _{{S}}(y^{2}z^{2}i+z^{2}x^{2}j+x^{2}y^{2}k)\cdot \,ds\, where S is the part of the sphere x^{2}+y^{2}+z^{2}=1\, above the xy-plane.

Solution If F=yi+(x-2xz)j-xyk\,,evaluate \iint _{{S}}(\nabla \times F)\cdot n\,dS\, where S is the surface of the sphere x^{2}+y^{2}+z^{2}=a^{2}\, above the xy-plane.

Solution Evaluate \iint _{{S}}({\mathrm  {curl}}F\cdot n\,dS\, where F=yi+zj+xk\, and surface S in the part of the sphere x^{2}+y^{2}+z^{2}=1\, above the xy-plane.

Solution Evaluate \iint _{S}(y^{2}zi+z^{2}xj+x^{2}yk)\cdot dS\, where S is the surface of the sphere x^{2}+y^{2}+z^{2}=a^{2}\, lying in the positive octant.

Solution Evaluate \iint _{{S}}F\cdot n\,dS\, over the surface of the cylinder x^{2}+y^{2}=9\, included in the first octant between z=0 and z=4 where F=zi+xj-yzk\,

Solution Evaluate \iint _{{S}}F\cdot n\,dS\, where F=2yxi-yzj+x^{2}k\, over the surface of the cube bounded by the coordinate planes and planes x=a,y=a,z=a\,

Solution Evaluate \iint _{{S}}F\cdot n\,dS\, where F=(x-z)i+(x^{3}+yz)j-3xy^{2}k\, and S in the surface of the cone z=2-{\sqrt  {(x^{2}+y^{2})}}\, above the xy-plane.

Solution Evaluate \iiint _{V}F\,dv\, where F=xi+yj+zk\, and V is the region bounded by the surfaces x=0,x=2,y=0,y=6,z=4,z=x^{2}\,

Solution Evaluate \iiint _{V}\phi \,dv\, where \phi =45x^{2}y\, and V is the closed region bounded by the planes 4x+2y+z=8,x=0,y=o,z=0\,

Solution If F=2xzi-xj+y^{2}k\, evaluate \iiint _{V}F\,dV\, where V is the region bounded by the planes x=y=z=0,x=y=z=1\,

Solution Let r denote the position vector any point (x,y,z) measured from an origin O and let r=|r|\,.Evaluate \iint _{S}{\frac  {r}{|r|^{3}}}\cdot dS\, where S denotes the sphere of radius a with center at the origin.

Solution Evaluate \iint _{S}F\cdot ndS\,,where F=yi+2xj-zk\, and S in the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4.

Solution If F=(2x^{2}-3z)i-2xyj-4xk\,,then evaluate i). \iiint _{V}\nabla \times FdV\, ii). \iiint _{V}FdV\,, where V is the region bounded by x=0,y=0,z=0 and 2x+2y+z=4\,

Solution Evaluate \iiint _{V}(2x+y)dV\, where V is closed region bounded by the cylinder z=4-x^{2}\, and the planes x=0,y=0,y=2 and z=0.

Solution Find the volume of the region common to the intersecting cylinders x^{2}+y^{2}=a^{2}\, and x^{2}+z^{2}=a^{2}\,

Green, Stokes & Gauss Divergence Theorems

  • Green's Theorem in the plane - Relation between plane and line integrals

If R is a closed region in the xy-plane bounded by a simple closed curve C and if \phi (x,y)\, and \psi (x,y)\, are continuous functions having continuous partial derivatives in R,then \oint (\psi dx+\phi dy)=\iint _{R}\left[{\frac  {\partial \phi }{\partial x}}-{\frac  {\partial \psi }{\partial y}}\right]dxdy\, where C is traversed in the positive (anti-clockwise) direction.

  • Stokes Theorem - Relation between surface and line integrals

If F is any continuously differentiable vector point function and S is a surface bounded by a curve C,then \oint F\cdot dr=\iint _{S}{\mathrm  {curl}}F\cdot ndS\, where the unit normal n at any point of S is drawn in the direction in which a right-handed screw would move when rotated in the sense of description of C.


Solution If F=(x^{2}-y^{2})i+2xyj\, and r=xi+yj\,,find the value of \int F\cdot dr\, around the rectangular boundary x=0,x=a,y=0,y=b.

SolutionEvaluate by Green's theorm in plane\int _{C}(e^{{-x}}\sin ydx+e^{{-x}}\cos ydy)\, where C is the rectangle with vertices (0,0),(\pi ,0),(\pi ,{\frac  {\pi }{2}}),(0,{\frac  {\pi }{2}})\,.

Solution Verify Green's theorm in plane for \int _{C}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]\, where C is the square with vertices (0,0),(2,0),(2,2),(0,2).

Solution Verify Green's theorm in plane for \oint _{C}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\, where C is the boundary of the region defined by x=0,y=0,x+y=1\,

Solution Apply Green's theorm in the plane to evaluate \int _{C}[(2x^{2}-y^{2})dx+(x^{2}+y^{2})dy]\, where C is the boundary of the curve enclosed by the x-axis and the semi-circley=(1-x^{2})^{{{\frac  {1}{2}}}}\,

Solution Show that the area bounded by a simple closed curve C is given by {\frac  {1}{2}}\int _{C}(xdy-ydx)\,. Hence deduce that the area of the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

Solution Verify Green's theorm in a plane \oint _{C}[(x^{2}-2xy)dx+(x^{2}y+3)dy]\, where C is the boundary of the region defined by y^{2}=8x,x=2\,

Solution Evaluate by Green's theorm \oint _{C}[(\cos x\sin y-xy)dx+\sin x\cos ydy]\, where C is the circle x^{2}+y^{2}=1\,

Solution Evaluate by Green's theorm in the plane \oint _{C}[(x^{2}-\cos hy)dx+(y+\sin x)dy]\, where C is the rectangle with vertices (0,0),(\pi \,,0),(\pi \,,1),(0,1).

Solution Evaluate \oint _{C}[(y-\sin x)dx+\cos xdy]\, where C is the triangle whose vertices are (0,0),({\frac  {\pi }{2}}\,,0),({\frac  {\pi }{2}}\,, by using Green's theorm in plane.

Solution Verify Green's theorm in the plane for \oint _{C}[(xy+y^{2})dx+x^{2}dy]\, where C is the closed curve of the region bounded by y=x,y=x^{2}\,

Solution Verify Green's theorm in plane for \oint _{C}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\, where C is the region bounded by the parabolas y^{2}=x,y=x^{2}\,

Solution Verify Stokes' theorm for F=(x^{2}+y^{2})i-2xyj\, taken round the rectangle bounded by x=\pm a,y=0,y=b\,

Solution Evaluate \oint _{C}F\cdot dr\, by Stokes' theorm where F=y^{2}i+x^{2}j-(x+z)k\, and C is the boundary of the triangle with vertices at (0,0,0),(1,0,0),(1,1,0).

Solution If F=(2x^{2}+y^{2})i+(3y-4x)j\, evaluate \oint _{C}F\cdot dr\, where C is the boundary of the triangle with vertices (0,0),(2,0),(2,1)

Solution Verify Stokes'theorm for the function F=x^{2}i+xyj\, integrated along the rectangle in the plane z=0,where sides are along the lines x=0,y=0,x=a and y=b.

Solution Evaluate by Stokes'theorm \oint _{C}(e^{x}dx+2ydy-dz)\, where C is the curve x^{2}+y^{2}=4,z=2\,

Solution Evaluate by Stokes'theorm \oint _{C}(\sin zdx-\cos xdy+\sin ydz)\, where C is the boundary of the rectangle 0\leq x\leq \pi ,0\leq y\leq 1,z=3\,

Solution By converting into line integral,evaluate \iint _{S}(\nabla \times A)\cdot ndS\,, where A=(x-z)i+(x^{3}+yz)j-3xy^{2}k\, and S is the surface of the cone z=2-{\sqrt  {(x^{2}+y^{2})}}\, above the xy-plane.

Solution By converting into a line integral evaluate \iint _{S}(\nabla \times F)\cdot ndS\, where F=(x^{2}+y-4)i+3xyj+(2xy+z^{2})k\, and S is the surface of the paraboloid z=4-(x^{2}+y^{2})\, above the xy-plane.

Solution Evaluate \iint _{S}(\nabla \times F)\cdot ndS\, where F=(y-z+2)i+(yz+4)j-xzk\, and S is the surface of the cube x=y=z=0,x=y=z=2\, above the xy-plane.

Solution Verify Stokes'theorm for F=yi+zj+xk\, where S is the upper half surface of the sphere x^{2}+y^{2}+z^{2}=1\, and C is its boundary.

Solution Verify Stokes'theorm for the vector F=3yi-xzj+yz^{2}k\, where S is the surface of the paraboloid 2z=x^{2}+y^{2}\, bounded by z=2 and C is its boundary.

Solution Verify Stokes'theorm for the function F=zi+xj+yk\, where C is the unit circle in xy-plane bounding the hemisphere z={\sqrt  {(1-x^{2}-y^{2})}}\,

Solution Apply Stokes'theorm to prove that \int _{C}(ydx+zdy+xdz)=-2{\sqrt  {2}}\pi a^{2}\,,where C is the curve given by x^{2}+y^{2}+z^{2}-2ax-2ay=0,x+y=2a\, and begins at the point (2a,0,0) and goes at first below the xy-plane.

Solution By Stokes' theorem,prove that {\mathrm  {curl}}{\mathrm  {grad}}\phi =0\,

Solution Evaluate \iint _{S}F\cdot ndS\, where F=axi+byj+czk\, and S is the surface of the sphere x^{2}+y^{2}+z^{2}=1\,

Solution Use divergence theorm to find \iint _{S}F\cdot ndS\, for the vector F=xi-yj+2zk\, over the sphere x^{2}+y^{2}+(z-1)^{2}=1\,

Solution If F=axi+byj+czk\, where a,b,c are constants,show that \iint _{S}(n\cdot F)dS={\frac  {4}{3}}\pi (a+b+c)\,,S being the surface of the sphere (x-1)^{2}+(y-2)^{2}+(z-3)^{2}=1\,

Solution Find \iint _{S}A\cdot ndS\,,where A=(2x+3z)i-(xz+y)j+(y^{2}+2z)k\, and S is the surface of the sphere having center at (3,-1,2) and radius 3 units.

Solution By using the Gauss Divergence theorm,evaluate \iint _{S}(xdydz+ydzdx+zdxdy)\,,where S is the surface of the sphere x^{2}+y^{2}+z^{2}=4\,

Solution Apply divergence theorm to evaluate \iint _{S}[(x+z)dydz+(y+z)dzdx+(x+y)dxdy]\, where S is the surface of the sphere x^{2}+y^{2}+z^{2}=4\,

Solution If F=4xzi-y^{2}j+yzk\,,then evaluate \iint _{s}F\cdot ndS\, where S is the surface of the cube enclosed by x=0,x=1,y=0,y=1,z=0 and z=1.

Solution Evaluate \iint F\cdot ndS\,,where F=4xyi+yzj-xzk\, and S is the surface of the cube bounded by the planes x=0,x=2,y=0,y=2,z=0,z=2.

Solution Apply Gauss'theorm to evaluate \iint _{S}[(x^{3}-yz)dzdx-2x^{2}ydzdx+zdxdy]\, over the surface S of a cube bounded by the coordinate planes and the planes x=y=z=a.

Solution Apply Gauss'theorm to show that \iint _{S}(x^{3}-yz)i-2x^{2}yj+2k]\cdot ndS={\frac  {a^{5}}{3}}\,,where S denotes the surface of the cube bounded by the planes x=0,x=a,y=0,y=a,z=0,z=a\,

Solution Evaluate \iint _{s}[x^{2}dydz+y^{2}dzdx+2z(xy-x-y)dxdy]\, where S is the surface of the cube 0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\,

Solution Find the value of \iint _{S}(F\times \nabla \phi )\cdot ndS\, where F=x^{2}i+y^{2}j+z^{2}k,\phi =xy+yz+zx,S:x=\pm 1,y=\pm 1,z=\pm 1\,

Solution Evaluate \iint _{S}F\cdot ndS\,,where F=xi-yj+(z^{2}-1)k\, and S is the closed surface bounded by the planes z=0,z=1\, and the cylinder x^{2}+y^{2}=4\, by the application of Gauss'theorm.

Solution Use Gauss' theorem to evaluate the integral \iint _{S}F\cdot ndS\, of the vector field F=xy^{2}i+y^{3}j+y^{2}zk\, through the closed surface formed by the cylinder x^{2}+y^{2}=9\, and the plane z=0,z=2\,

Solution Use Gauss divergence theorem to find \iint _{S}F\cdot ndS\, where F=2x^{2}yi-y^{2}j+4xz^{2}k\, and S is the closed surface in the first octant bounded by y^{2}+z^{2}=9,x=2\,

Solution Evaluate \iint _{S}(zx^{2}dxdy+x^{3}dydz+yx^{2}dzdx)\, where S is the closed surface consisting of the cylinder x^{2}+y^{2}=4\, and the circular discs z=0,z=3\,

Solution If F=\nabla \phi ,\nabla ^{2}\phi =-4\pi \rho \, show that \iint _{S}F\cdot ndS=-4\pi \iiint _{V}\rho dV\,

SolutionIf \phi \, is harmonic in V,then \iint _{S}{\frac  {\partial \phi }{\partial n}}dS=\iiint _{V}\nabla ^{2}\phi dV\, where S is the surface enclosing V.

Solution Prove that i). \iiint _{V}\nabla \phi \cdot AdV=\iint _{S}\phi \cdot ndS-\iiint _{V}\phi \nabla \cdot AdV\, ii).Prove that \iiint _{V}F\cdot {\mathrm  {curl}}GdV=\iint _{S}G\times F\cdot dS+\iiint _{V}G\cdot {\mathrm  {curl}}FdV\,

Solution Show that for any closed surface S, i). \iint _{S}ndS=0\, ii).\iint _{S}r\times ndS=0\, iii).\iint _{S}(\nabla \phi )\times ndS=0\,

Solution If V is the volume of a region T bounded by a surface S,then prove that V=\iint _{S}xdydz=\iint _{S}ydzdx=\iint _{S}zdxdy\,

Solution Evaluate \iint _{S}(\nabla \times F)\cdot ndS\,,where F=(x-z)i+(x^{3}+yz)j-3xy^{2}k\, and S is the surface of the cone z=2-{\sqrt  {(x^{2}+y^{2})}}\, above the xy plane.

Solution Evaluate \iint _{S}(x^{3}dydz+y^{3}dzdx+z^{3}dxdy)\, by converting the surface integral into a volume integral.Here,S is the surface of the sphere x^{2}+y^{2}+z^{2}=1\,

Solution Evaluate with the help of divergence theorm the integral \iint _{S}[xz^{2}dydz+(x^{2}y-z^{3})dzdx+(2xy+y^{2}z)dxdy]\,, where S is the entire surface of the hemispherical region bounded by z={\sqrt  {(a^{2}-x^{2}-y^{2})}},z=0\,

Solution Evaluate \iint _{S}(ax^{2}+by^{2}+cz^{2})dS\, over the sphere x^{2}+y^{2}+z^{2}=1\, using the divergence theorm.

Solution Compute \iint _{S}(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2})^{{{\frac  {1}{2}}}}dS\, over the ellipsoid ax^{2}+by^{2}+cz^{2}=1\,

Solution Compute \iint _{S}(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2})^{{{\frac  {-1}{2}}}}dS\, over the ellipsoid ax^{2}+by^{2}+cz^{2}=1\,

Solution Evaluate\iint _{S}F\cdot ndS\, over the entire surface of the region above the xy plane bounded by the cone z^{2}=x^{2}+y^{2}\, and the plane z=4,if F=xi+yj+z^{2}k\,

Solution By using the Gauss divergence theorm evaluate \iint _{S}(xi+yj+z^{2}k)\cdot ndS\,,where S is the closed surface bounded by the cone x^{2}+y^{2}=z^{2}\, and the plane z=1.

Solution Evaluate \iint _{S}(\nabla \times F)\cdot ndS\, where A=[xye^{z}+\log(z+1)-\sin x]k\, and S is the surface of the sphere x^{2}+y^{2}+z^{2}=a^{2}\, above the xy plane.

Solution Evaluate \iint _{S}xyzdS\, where S is the surface of the sphere x^{2}+y^{2}+z^{2}=a^{2}\,

Solution By transforming to a triple integral,evaluate \iint _{S}(x^{3}dydz+x^{2}ydzdx+x^{2}zdxdy)\, where S is the closed surface bounded by the planes z=0,z=b\, and the cylinder x^{2}+y^{2}=a^{2}\,

Solution Evaluate \iint _{S}{\frac  {1}{p}}dS\, where S is the surface of the ellipsoid {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}+{\frac  {z^{2}}{c^{2}}}=1\, and p is the perpendicular drawn from the origin to the tangent plane at (x,y,z)\,

Solution Show that \iint _{S}(x^{2}i+y^{2}j+z^{2}k)\cdot ndS\, vanishes where S denotes the surface of the ellipsoid {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}+{\frac  {z^{2}}{c^{2}}}=1\,

Solution Verify the divergence theorm theorm for F=4xzi-y^{2}j+yzk\, taken over the cube bounded by x=0,x=1,y=0,y=1,z=0,z=1\,

Solution Evaluate \iint _{S}(x^{3}dydz+y^{3}dzdx)\, where S is the surface of hte sphere x^{2}+y^{2}+z^{2}=a^{2}\,

Solution Show that the vector field F=(2xy^{2}+yz)i+(2x^{2}y+xz+2yz^{2})j+(2y^{2}z+xy)k\, is conservative.

Solution Show that the vector field defined by F=(2xy-z^{3})i+(x^{2}+z)j+(y-3xz^{2})k\, is conservative and find the scalar potential of F.

Solution Show that the vector field F given by F=(y+\sin z)i+xj+x\cos zk\, is conservative,find its scalar potential.

Solution Show that F=xi+yj+zk\, is conservative and find \phi \, such that F=\nabla \phi \,

Solution Prove that F=|r|^{2}r\, is conservative and find its scalar potential.

Solution Show that (y^{2}z^{3}\cos x-4x^{3}z)dx+2z^{3}y\sin xdy+(3y^{2}z^{2}\sin x-x^{4})dz\, is an exact differential of some function \phi \, and find this function.

Solution Show that (2x\cos y+z\sin y)dx+(xz\cos y-x^{2}\sin y)dy+x\sin ydz=0\, is an exact differential and hence solve it.

Solution Evaluate \int _{C}[2xyz^{2}dx+(x^{2}z^{2}+z\cos yz)dy+(2x^{2}yz+y\cos yz)dz\, where C is any path from (0,0,0)\, to 1,{\frac  {\pi }{4}},2)\,

Solution If F=\cos yi-x\sin yj\,, evaluate \int _{C}F\cdot dr\, where C is the curve y={\sqrt  {(1-x^{2})}}\, in the xy plane from (1,0)\, to (0,1)\,

Solution Evaluate \int _{C}[yzdx+(xz+1)dy+xydz]\,,where C is any path from (1,0,0)\, to (2,1,4)\,

Solution A vector field is given by F=(x^{2}+xy^{2})i+(y^{2}+x^{2}y)j\, Show that the field is irrational and obtain its scalar potential.

Solution Show that the vector field F given by F=(x^{2}-yz)i+(y^{2}-zx)j+(z^{2}-xy)k\, is irrotational. Find a scalar \phi \, such that F=\nabla \phi \,

Solution Show that the vector function F=(\sin y+z\cos x)i+(x\cos y+\sin z)j+(y\cos z+\sin x)k\, is irrotational and find the scalar function \phi \, such that F=\nabla \phi \,

Multiple Integrals

solution \int _{0}^{2}\int _{0}^{1}(2x+y)^{8}dxdy\,

solution Evaluate \int _{0}^{2}\int _{1}^{2}(x^{2}+y^{2})dxdy\,

solution Evaluate \int _{0}^{1}\int _{1}^{2}(x^{2}+y^{2})dxdy\,

solution \int _{0}^{3}\int _{1}^{2}xy(x+y)dxdy\,

solution \int _{0}^{a}\int _{0}^{b}(x^{2}+y^{2})dxdy\,

solution \int _{1}^{2}\int _{3}^{4}{\frac  {1}{(x+y)^{2}}}dxdy\,

solution \int _{1}^{4}\int _{{0}}^{{{\sqrt  {4-x}}}}xydxdy\,

solution \int _{1}^{2}\int _{{x}}^{{x{\sqrt  {3}}}}xydxdy\,

solution \int _{1}^{2}\int _{1}^{x}xy^{2}dxdy\,

solution \int _{{0}}^{{{\frac  {\pi }{4}}}}\int _{{0}}^{{{\frac  {\pi }{2}}}}\sin(x+y)dxdy\,

solution \int _{0}^{a}\int _{{0}}^{{{\sqrt  {a^{2}-x^{2}}}}}y^{3}dydx\,

solution \int _{0}^{1}\int _{{{\sqrt  {y}}}}^{{2-y}}x^{2}dxdy\,

solution \int _{0}^{2}\int _{{x^{2}}}^{{2x}}(2x+3y)dydx\,

solution \int _{0}^{a}\int _{{0}}^{{{\sqrt  {a^{2}-x^{2}}}}}xydxdy\,

solution \int _{0}^{1}\int _{{0}}^{{1-x}}(x^{2}+y^{2})dydx\,

solution \int _{0}^{a}\int _{{{\frac  {x^{2}}{a}}}}^{{2a-x}}xydydx\,

solution \int _{0}^{1}\int _{{-{\sqrt  {y}}}}^{{{\sqrt  {y}}}}dxdy+\int _{1}^{9}\int _{{{\frac  {y-3}{2}}}}^{{{\sqrt  {y}}}}dxdy\,

solution \int _{{0}}^{{2a}}\int _{{{\frac  {y^{2}}{4a}}}}^{{3a-y}}(x^{2}+y^{2})dxdy\,

solution Evaluate \iint _{R}xydxdy\, where R is the positive quadrant of the circle x^{2}+y^{2}=a^{2}\,

solution Evaluate the double integral \iint xy(x+y)dxdy\, over the region bounded by the curves y=x,y=x^{2}\,

solution Evaluate \int _{{0}}^{{2a}}\int _{{0}}^{{{\sqrt  {2ax-x^{2}}}}}(x^{2}+y^{2})dxdy\, by changing into polar coordinates.

solution Evaluate \iint xy(x^{2}+y^{2})^{{{\frac  {n}{2}}}}dxdy\, over the positive quadrant of the circle x^{2}+y^{2}=a^{2}\, supposing n+3>0.

solution Find \iint _{R}xydxdy\, where R is the region bounded by x=1,x=2,y=0,xy=1\,

solution Evaluate \int _{{1}}^{{\log 8}}\int _{{0}}^{{\log y}}e^{{x+y}}dxdy\,

solution Evaluate I=\iint _{D}(x^{2}+y^{2})dxdy\, where D is bounded by y=x\, and y^{2}=4x\,

solution \iint _{D}(4xy-y^{2})dxdy\, where D is the reactangle bounded by x=1,x=2,y=0,y=3\,

solution \iint _{D}(x^{2}+y^{2})dxdy\, where D is the region bounded by y=x,y=2x,x=1\, in the first quadrant.

solution \iint _{D}(1+x+y)dxdy\, where D is the region bounded by the lines y=-x,x={\sqrt  {y}},y=2,y=0\,

solution \iint _{D}xydxdy\, where D is the domain bounded by the parabola x^{2}=4ay\,,the ordinates x=a\, and x-axis.

solution Evaluate \iint (x-y)dxdy\, over the region between the line x=y\, and the parabola y=x^{2}\,

solution Find the area bounded by the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

solution I_{D}=\iint _{D}x^{3}ydxdy\, where D is the region enclosed by the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\, in the first quadrant.

solution Find by double integration,the area which lies inside the cardoid r=a(1+\cos \theta )\, and outside the circle r=a.

solution Find the area in the XY-plane bounded by the lemniscate r^{2}=a^{2}\cos 2\theta \,

solution Find the area bounded by the curves y^{2}=x^{3}\, and x^{2}=y^{3}\,

solution Find the area of the domains 3x=4-y^{2},x=y^{2}\, in the XY-plane.

solution Find the area of the region bounded by the parabola y^{2}=4ax\, and the straight line x+y=3a\, in the XY-plane.

solution Find the area common to the parabolas y^{2}=4a(x+a),y^{2}=4b(b-x)\,

solution Find the area of the domains x=y-y^{2},x+y=0\, in the XY-plane.

solution Find the area of the domains 3y^{2}=25x,5x^{2}=9y\, in the XY-plane.

solution Find the mass,coordinates of the centre of gravity and moments of inertia relative to x-axis,y-axis and origin of a reactangle 0\leq x\leq 4,0\leq y\leq 2\, having mass density xy.

solution Find the volume of tetrahedron in space cut from the first octant by the plane 6x+3y+2z=6\,

solution Calculate the volume of a solid whose base is in a xy-plane and is bounded by the parabola y=4-x^{2}\, and the straight line y=3x\,,while the top of the solid is in the plane z=x+4\,.

solution Find the moment of inertia of the area bounded by the circle x^{2}+y^{2}=a^{2}\, in the first quadrant,assume the surface density of 1.

solution A plane lamina of non uniform density is in the form of a quadrant of the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,.If the density at any point (x,y)be kxy,where K is a constant,find the coordinates of the centroid of the lamina.

solution Determine the volume of the space below the paraboloid x^{2}+y^{2}+z-4=0\, and above the square in the xy-plane with vertices at (0,0),(0,1),(1,0),(1,1)\,.

solution Find the volume of the solid under the surface az=x^{2}+y^{2}\, and whose base R is the circle x^{2}+y^{2}=a^{2}\,.

solution Find the volume enclosed by the coordinate planes and that portion of the plane x+y+z=1\, which lies in the first quadrant.

solution A circular hole of radius b is made centrally through a sphere of radius a.Find the volume of the remaining portion of the sphere.

solution Find the volume of the region bounded by the paraboloids z=x^{2}+y^{2}\, and z={\frac  {6-{\frac  {x^{2}+y^{2}}{2}}}\,}.

solution Find the volume of the region bounded by the paraboloid z=x^{2}+y^{2}\, and the plane z=4.

solution Find the volume of the solid enclosed by the ellipsoid {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}+{\frac  {z^{2}}{c^{2}}}=1\,

solutionFind the volume of the region in space bounded by the surface z=1-(x^{2}+y^{2})\, on the sides by the planes x=0,y=0,x+y=1\, and below by the plane z=0.

solution Evaluate \int _{0}^{1}\int _{0}^{x}\int _{{0}}^{{x+y}}(x+y+z)dzdydx\,

solution Find the volume bounded by the ellipsoidic paraboloids z=x^{2}+9y^{2}\, and z=18-x^{2}-9y^{2}\,

solution Find the total mass of the region in the cube 0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\, with density at any point given by xyz\,

solution Find the mass,centroid of the tetrahedron bounded by the coordinate planes and the plane {\frac  {x}{a}}+{\frac  {y}{b}}+{\frac  {z}{c}}=1\,

solution Evaluate \int _{0}^{2}\int _{1}^{z}\int _{{0}}^{{yz}}xyzdxdydz\,

solution If the radius of the base and altitude of a right circular cone are given by a and h respectively,express its volume as a tripple integral and evaluate it using cylindrical coordinates.

solution Evaluate \int _{0}^{a}\int _{0}^{x}\int _{{0}}^{{y+x}}e^{{x+y+z}}dzdydx\,

solution Find the volume bounded by the sphere x^{2}+y^{2}+z^{2}=a^{2}\,
solution Evaluate \iiint xyzdxdydz\, over the positive octant of the sphere x^{2}+y^{2}+z^{2}=a^{2}\,

solution Assuming \rho (x,y,z)=1\,,find the centroid of the portion of the sphere x^{2}+y^{2}+z^{2}=a^{2}\, in the first octant.

solution Find the mass and moment of inertia of a sphere of radius 'a' with respect to a diameter if the density is proportional to the distance from the center.

solution Evaluate \iiint _{E}y^{2}x^{2}dV\, Where E is the region bounded by the paraboloid x=1-y^{2}-z^{2}\, and the plane \,x=0

solution \int _{{-2}}^{{2}}\int _{{0}}^{{{\sqrt  {4-y^{2}}}}}\int _{{-{\sqrt  {4-x^{2}-y^{2}}}}}^{{{\sqrt  {4-x^{2}-y^{2}}}}}y^{2}{\sqrt  {x^{2}+y^{2}+z^{2}}}dzdxdy\, Evaluate using spherical coordinates

solution Evaluate \iint _{{R}}(x+y)e^{{x^{2}-y^{2}}}dA\, Where _{{R}}\, is the rectangle enclosed by the lines x-y=0,x-y=2,x+y=0,x+y=3\,