Scalar and Vector Fields
We will cover following topics
Scalar Fields
If to each point P(→r) (the point P with position vector →r ) of a region R in space there is a unique scalar or real number denoted by ϕ(→r), then ϕ is called a scalar point function in R. The region R is called a scalar field.
Vector Fields
If to each point P(→r) of a region R in space there is a unique vector denoted by →F(→r) then →F is called a vector point function in R. The region R is called a vector field.
Derivative of a Vector Function
A vector function →f(t) is said to be differentiable at a point t, if limΔt→0→f(t+Δt)−→f(t)Δt exists. Then it is denoted by d→fdt or →f′ and is called the derivative of the vector function →f at t.
Geometrical Meaning of Derivative
Let →r(t) be the position vector of a point P with respect to the origin O. As t varies continuously over a time interval P traces the curve C. Thus, the vector function →r(t) represents a curve C in space.
If limΔ→0Δ→rΔt exists, it is denoted by d→rdt and d→rdt is in the direction of the tangent at P to the curve.
If d→rdt≠0, then d→rdt or →r′(t) is called a tangent vector to the curve C at P.
| The unit tangent vector at P is $$=\dfrac{\vec{r}^{\prime}(t)}{\left | \vec{r}^{\prime}(t)\right | }=\hat{u}(t)$$ |
Both →r′(t) and ˆu(t) are in the direction of increasing t. Hence, their sense depends on the orientation of the curve C.
Level Surface
Let ϕ be a continuous scalar point function defined in a region R in space. Then the set of all points satisfying the equation ϕ(x,y,z)=C, where C is a constant, determines a surface which is called a level surface of ϕ.
PYQs
Vector Fields
1) Evaluate ∫c→A⋅d→r along the curve x2+y2=1, z=1 from (0,1,1) to (1,0,1) if →A=(yz+2x)ˆi+xzˆj+(xy+2z)ˆk.
[2008, 15M]
2) Evaluate ∬, where \vec{\mathrm{F}}=4 x \hat{i} -2y^2 \hat{j}+ z^2 \hat{k}, and S is the surface of the cylinder bounded by x^{2}+y^{2}=4, z=0 and z=3.
[2008, 15M]
Differentiation of a Vector Field of a Scalar Variable
1) The position vector of a moving point at time t is \overline{r}=\sin t \hat{i}+\cos 2 t \hat{j}+\left(t^{2}+2 t\right) \hat{k}. Find the components of acceleration \overline{a} in the direction parallel to the velocity vector \overline{v} and perpendicular to the plane of \overline{r} and \overline{v} at time t=0.
[2017, 10M]
2) For two vectors \vec{a} and \vec{b} given respectively by \vec{a}=5 t^{2} \hat{i}+\hat{t} \hat{\jmath}-t^{3} \hat{k} and \vec{b}=\sin 5 t \hat{i}-\cos t \hat{j}, determine:
i) \dfrac{d}{d t}(\vec{a} \cdot \vec{b})
ii) \dfrac{d}{d t}(\vec{a} \times \vec{b})
[2011, 20M]