# Summary

• A quantity that has only magnitude is said to be a scalar quantity, such as time, mass, distance, temperature, work, electric potential, etc. A quantity that has both magnitude and direction is called a vector quantity, such as force, velocity, displacement, electric field intensity, etc.
• If the value of the physical function at each point is a scalar quantity then the field is known as a scalar field, such as temperature distribution in a building. If the value of the physical function at each point is a vector quantity then the field is known as a vector field, such as the gravitational force on a body in space.
• A unit vector  along  is defined as a vector whose magnitude is unity and its direction is along  In general, any vector can be represented as

where A or  represents the magnitude of the vector and , direction of the vector .
• Two vectors can be added together by the triangle rule or parallelogram rule of vector addition.
• The dot product of two vectors  and , written as , is defined as,
where  is the smaller angle between  and , and  and  represent the magnitude of  and , respectively.
• The cross product of two vectors  and , written as , is defined as,

where  is the unit vector normal to the plane containing  and . The direction of the cross product is obtained from a common rule, called right-hand rule.
• ​Three orthogonal coordinate systems commonly used are Cartesian coordinates (x, y, z), cylindrical coordinates (r, φ, z) and spherical coordinates (ρθφ).
• The differential lengths in three coordinate systems are given respectively as,

• The differential areas in three coordinate systems are given respectively as,

• The differential volumes in three coordinate systems are given respectively as,

• For the vector  and a path l, the line integral is given by,
• If the path of integration is a closed curve, the line integral is the circulation of the vector around the path.
• If the line integration of a vector along a closed path is zero, i.e.,  then the vector is known as conservative or lamellar.
• For a vector , continuous in a region containing a smooth surface S, the surface integral or the flux of  through S is defined as,
where,  is the unit normal vector to the surface S.
• If the surface is a closed surface, the surface integral is the net outward flux of the vector.
• If the surface integral of a vector over a closed surface is zero, i.e.,  then the vector is known as a solenoidal vector.
• The volume integral of a scalar quantity F over a volume V is written as,
• The differential vector operator () or del or nabla, defined in Cartesian coordinates as,
​​
is merely a vector operator, but not a vector quantity. It is a vector space function operator. It is used for performing vector differentiations.
• The gradient of a scalar function is both the magnitude and the direction of the maximum space rate of change of that function.
• The gradient of a scalar quantity in three different coordinate systems is expressed respectively as,
• The divergence of a three-dimensional vector field at a point is a measure of how much the vector diverges or converges from that point.
• If the divergence of a vector is zero then the vector is known as a solenoidal vector.
• The divergence of a vector quantity in three different coordinate systems is expressed respectively as,
• The curl of a vector field, denoted by curl  or , is defined as the vector field having magnitude equal to the maximum circulation at each point and to be oriented perpendicularly to this plane of circulation for each point.
• If the curl of a vector is zero then the vector is known as irrotational vector.
• The curl of a vector quantity in three different coordinate systems is expressed respectively as,
• The Laplacian operator (2) of a scalar field is the divergence of the gradient of the scalar field upon which the operator operates.
• The Laplacian operator (2) of a scalar field in three different coordinate systems is expressed respectively as,
• Gauss’ divergence theorem is used to convert a volume integral into surface integral and vice versa. According to this theorem, the divergence of a vector field over a volume is equal to the surface integral of the normal component of the vector through the closed surface bounding the volume.
where V is the volume enclosed by the closed surface S.
• According Green’s theorem, if  is a two-dimensional vector field, such that then,

where C is a positively oriented boundary of the region R.
• Stokes’ theorem is used to convert a line integral into surface integral and vice versa. According to this theorem, the line integral of a vector around a closed path is equal to the surface integral of the normal component of is curl over the surface bounded by the path.
where S is the surface enclosed by the path L. The positive direction of  is related to the positive sense of defining L according to the right-hand rule.
• The Helmholtz theorem states that a vector field is uniquely described within a region by its divergence and curl.