Multivariable Calculus
Contents
Vector Calculus
Vector Differentiation
Solution If calculate at the point (1,1,0).
Solution Find when
Solution If Find
Solution If where n is constant and t varies, prove that and .
Solution If where a,b are constant vectors, show that
Solution If ,Show that and
Solution If and , Find and where t=1.
Solution If .Find at
Solution A particle moves along a curve whose parametric equations are .Find the velocity and acceleration at t=0.
Solution A particle moves along the curve where t is the time.Find the components of its velocity and acceleration at t=1 in the direction of .
Solution A particle moves so that its position vector is given by where 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 is the path of a particle moving with constant acceleration.
SolutionIf ,find the value of .
Solution If .Verify that .
Solution If and .Find at (2,-1,1).
Vector Integration
Solution If ,Find i). ii).
Solution If ,find .
Solution Evaluate
Solution If and ,Evaluate and .
Solution Evaluate where
Solution Given that when t=2, when t=3, Show that .
Solution Evaluate where
Solution If ,prove that
Solution Evaluate
Solution Evaluate where
Solution The acceleration of a moving particle at any time t is given by .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 given that at t=0.
Solution If the acceleration of a particle at any time t greater than or equal to zero is given by and the velocity v and displacement r are zero at t=0, then find v and r at any time t.
Solution Integrate
Gradient Divergence and Curl
SolutionIf then find
SolutionIf ,find at the point (1,-2,-1).
Solution If and , Prove that
i). ii).
Solution If ,Show that
Solution If Prove that
Solution Evaluate where
Solution Show that
Solution If ,Prove that i). ii). iii).
SolutionIf ,find
SolutionFind where
SolutionIf , find i). . ii). Evaluate
Solution If ,prove that
SolutionIf ,find
Solution Evaluate where a is a constant vector.
Solution Find or curl F, where
i). ii).
Solution Prove that where
Solution If ,find
Solution If ,prove that i).
ii). If show that
Solution IF ,Find .What are their values at
Solution Find the of the vector at the point
Solution If ,prove that
Solution a). Prove that vector is solenoidal.
b). Determine the constant 'a' so that the vector is solenoidal.
Solution a). Show that the vector is irrational. b). Determine the constants 'a','b','c' so that the vector is irrational.
Solution Prove that
Solution Prove or
Solution If a is a constant vector,prove that
Solution Show that
Solution Show that
Solution Evaluate where
Solution If ,Show that
Solution Show that satisfies Laplace equation
Solution If f and g are two scalar functions,prove that
Solution Show that if
Solution Prove that ,where
Solution Evaluate
Solution Prove that
SolutionIf ,prove that
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 satisfies the equation
Solution Prove that
Solution Prove that
Solution If and ,find at the point (1,0,2).
Solution Prove that ,where a is a constant vector.
Solution Find the unit normal to the surface at the point (-1,-2,5).
Solution Find the directional derivative of at (1,-2,-1) in the direction of
Solution Calculate the maximum rate of change and the corresponding direction for the function at the point
Solution Find the equation of the tangent plane and normal to the surface at the point (1,2,2).
Solution Find the equation of the tangent line and normal plane to the curve of intersection of at (1,0,0).
Solution Find the angle between the curves at the point (2,-1,2).
Solution Find the constants a and b so that surfaces will be orthogonal to the surface at the point (1,-1,2).
Line, Surface & Volume Integrals
Solution If ,Evaluate ,where C is the curve in the xy-plane from (0,0)to (1,2).
Solution Evaluate where ,C is the rectangle in xy-plane bounded by
Solution If ,evaluate where C is the curve in the xy-plane consisting of the straight line from to and then to
Solution If ,evaluate where C is the straight line joining
Solution Evaluate where and C is the curve in the xy-plane from to
Solution Evaluate along the curve in the positive direction from (0,1,1) to (1,0,1),where
Solution Evaluate where and the curve c is in the xy-plane from (0,0) to (4,4) where
Solution Evaluate where and C in the arc of the curve from (2,0) to (4,12).
Solution If evaluate around the triangle ABC whose vertices are
Solution If .evaluate 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 ,evaluate along the curve C. from t=0 to t=1.
Solution Evaluate where and C is the arc of the curve from to
Solution Evaluate where and C is the portion of the curve from to
Solution Evaluate where and C is the arc of the curve from to
Solution If ,evaluate where C is the curve from t=0 to 1.
Solution Find the total work done in moving a particle in a force field given by along the curve from t=1 to t=2.
Solution Find the work done when a force moves a particle in xy-plane from (0,0) to (1,1) along the curve
Solution Find the work done in moving a particle in a force field 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
Solution Find the circulation of F round the curve C where and C is the circle
Solution Find the circulation of F round the curve C where and C is the rectangle whose vertices are
Solution Evaluate where S is the part of the sphere above the xy-plane.
Solution If ,evaluate where S is the surface of the sphere above the xy-plane.
Solution Evaluate where and surface S in the part of the sphere above the xy-plane.
Solution Evaluate where S is the surface of the sphere lying in the positive octant.
Solution Evaluate over the surface of the cylinder included in the first octant between z=0 and z=4 where
Solution Evaluate where over the surface of the cube bounded by the coordinate planes and planes
Solution Evaluate where and S in the surface of the cone above the xy-plane.
Solution Evaluate where and V is the region bounded by the surfaces
Solution Evaluate where and V is the closed region bounded by the planes
Solution If evaluate where V is the region bounded by the planes
Solution Let r denote the position vector any point (x,y,z) measured from an origin O and let .Evaluate where S denotes the sphere of radius a with center at the origin.
Solution Evaluate ,where and S in the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4.
Solution If ,then evaluate i). ii). , where V is the region bounded by x=0,y=0,z=0 and
Solution Evaluate where V is closed region bounded by the cylinder and the planes x=0,y=0,y=2 and z=0.
Solution Find the volume of the region common to the intersecting cylinders and
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 and are continuous functions having continuous partial derivatives in R,then 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 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 and ,find the value of around the rectangular boundary x=0,x=a,y=0,y=b.
SolutionEvaluate by Green's theorm in plane where C is the rectangle with vertices (0,0),.
Solution Verify Green's theorm in plane for where C is the square with vertices (0,0),(2,0),(2,2),(0,2).
Solution Verify Green's theorm in plane for where C is the boundary of the region defined by x=0,y=0,
Solution Apply Green's theorm in the plane to evaluate where C is the boundary of the curve enclosed by the x-axis and the semi-circle
Solution Show that the area bounded by a simple closed curve C is given by . Hence deduce that the area of the ellipse
Solution Verify Green's theorm in a plane where C is the boundary of the region defined by
Solution Evaluate by Green's theorm where C is the circle
Solution Evaluate by Green's theorm in the plane where C is the rectangle with vertices (0,0),(,0),(,1),(0,1).
Solution Evaluate where C is the triangle whose vertices are (0,0),(,0),(, by using Green's theorm in plane.
Solution Verify Green's theorm in the plane for where C is the closed curve of the region bounded by
Solution Verify Green's theorm in plane for where C is the region bounded by the parabolas
Solution Verify Stokes' theorm for taken round the rectangle bounded by
Solution Evaluate by Stokes' theorm where and C is the boundary of the triangle with vertices at (0,0,0),(1,0,0),(1,1,0).
Solution If evaluate where C is the boundary of the triangle with vertices (0,0),(2,0),(2,1)
Solution Verify Stokes'theorm for the function 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 where C is the curve
Solution Evaluate by Stokes'theorm where C is the boundary of the rectangle
Solution By converting into line integral,evaluate , where and S is the surface of the cone above the xy-plane.
Solution By converting into a line integral evaluate where and S is the surface of the paraboloid above the xy-plane.
Solution Evaluate where and S is the surface of the cube above the xy-plane.
Solution Verify Stokes'theorm for where S is the upper half surface of the sphere and C is its boundary.
Solution Verify Stokes'theorm for the vector where S is the surface of the paraboloid bounded by z=2 and C is its boundary.
Solution Verify Stokes'theorm for the function where C is the unit circle in xy-plane bounding the hemisphere
Solution Apply Stokes'theorm to prove that ,where C is the curve given by and begins at the point (2a,0,0) and goes at first below the xy-plane.
Solution By Stokes' theorem,prove that
Solution Evaluate where and S is the surface of the sphere
Solution Use divergence theorm to find for the vector over the sphere
Solution If where a,b,c are constants,show that ,S being the surface of the sphere
Solution Find ,where 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 ,where S is the surface of the sphere
Solution Apply divergence theorm to evaluate where S is the surface of the sphere
Solution If ,then evaluate 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 ,where 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 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 ,where S denotes the surface of the cube bounded by the planes
Solution Evaluate where S is the surface of the cube
Solution Find the value of where
Solution Evaluate ,where and S is the closed surface bounded by the planes and the cylinder by the application of Gauss'theorm.
Solution Use Gauss' theorem to evaluate the integral of the vector field through the closed surface formed by the cylinder and the plane
Solution Use Gauss divergence theorem to find where and S is the closed surface in the first octant bounded by
Solution Evaluate where S is the closed surface consisting of the cylinder and the circular discs
Solution If show that
SolutionIf is harmonic in V,then where S is the surface enclosing V.
Solution Prove that i). ii).Prove that
Solution Show that for any closed surface S, i). ii). iii).
Solution If V is the volume of a region T bounded by a surface S,then prove that
Solution Evaluate ,where and S is the surface of the cone above the xy plane.
Solution Evaluate by converting the surface integral into a volume integral.Here,S is the surface of the sphere
Solution Evaluate with the help of divergence theorm the integral , where S is the entire surface of the hemispherical region bounded by
Solution Evaluate over the sphere using the divergence theorm.
Solution Compute over the ellipsoid
Solution Compute over the ellipsoid
Solution Evaluate over the entire surface of the region above the xy plane bounded by the cone and the plane z=4,if
Solution By using the Gauss divergence theorm evaluate ,where S is the closed surface bounded by the cone and the plane z=1.
Solution Evaluate where and S is the surface of the sphere above the xy plane.
Solution Evaluate where S is the surface of the sphere
Solution By transforming to a triple integral,evaluate where S is the closed surface bounded by the planes and the cylinder
Solution Evaluate where S is the surface of the ellipsoid and p is the perpendicular drawn from the origin to the tangent plane at
Solution Show that vanishes where S denotes the surface of the ellipsoid
Solution Verify the divergence theorm theorm for taken over the cube bounded by
Solution Evaluate where S is the surface of hte sphere
Solution Show that the vector field is conservative.
Solution Show that the vector field defined by is conservative and find the scalar potential of F.
Solution Show that the vector field F given by is conservative,find its scalar potential.
Solution Show that is conservative and find such that
Solution Prove that is conservative and find its scalar potential.
Solution Show that is an exact differential of some function and find this function.
Solution Show that is an exact differential and hence solve it.
Solution Evaluate where C is any path from to
Solution If , evaluate where C is the curve in the xy plane from to
Solution Evaluate ,where C is any path from to
Solution A vector field is given by Show that the field is irrational and obtain its scalar potential.
Solution Show that the vector field F given by is irrotational. Find a scalar such that
Solution Show that the vector function is irrotational and find the scalar function such that
Multiple Integrals
solution Evaluate
solution Evaluate
solution Evaluate where R is the positive quadrant of the circle
solution Evaluate the double integral over the region bounded by the curves
solution Evaluate by changing into polar coordinates.
solution Evaluate over the positive quadrant of the circle supposing n+3>0.
solution Find where R is the region bounded by
solution Evaluate
solution Evaluate where D is bounded by and
solution where D is the reactangle bounded by
solution where D is the region bounded by in the first quadrant.
solution where D is the region bounded by the lines
solution where D is the domain bounded by the parabola ,the ordinates and x-axis.
solution Evaluate over the region between the line and the parabola
solution Find the area bounded by the ellipse
solution where D is the region enclosed by the ellipse in the first quadrant.
solution Find by double integration,the area which lies inside the cardoid and outside the circle r=a.
solution Find the area in the XY-plane bounded by the lemniscate
solution Find the area bounded by the curves and
solution Find the area of the domains in the XY-plane.
solution Find the area of the region bounded by the parabola and the straight line in the XY-plane.
solution Find the area common to the parabolas
solution Find the area of the domains in the XY-plane.
solution Find the area of the domains 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 having mass density xy.
solution Find the volume of tetrahedron in space cut from the first octant by the plane
solution Calculate the volume of a solid whose base is in a xy-plane and is bounded by the parabola and the straight line ,while the top of the solid is in the plane .
solution Find the moment of inertia of the area bounded by the circle 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 .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 and above the square in the xy-plane with vertices at .
solution Find the volume of the solid under the surface and whose base R is the circle .
solution Find the volume enclosed by the coordinate planes and that portion of the plane 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 and .
solution Find the volume of the region bounded by the paraboloid and the plane z=4.
solution Find the volume of the solid enclosed by the ellipsoid
solutionFind the volume of the region in space bounded by the surface on the sides by the planes and below by the plane z=0.
solution Evaluate
solution Find the volume bounded by the ellipsoidic paraboloids and
solution Find the total mass of the region in the cube with density at any point given by
solution Find the mass,centroid of the tetrahedron bounded by the coordinate planes and the plane
solution Evaluate
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
solution Find the volume bounded by the sphere
solution Evaluate over the positive octant of the sphere
solution Assuming ,find the centroid of the portion of the sphere 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 Where E is the region bounded by the paraboloid and the plane
solution Evaluate using spherical coordinates
solution Evaluate Where is the rectangle enclosed by the lines