Vector Identities

\(\nabla\) operator applied once to point functions

i) If \(f\) is a scalar point function and \(\vec{G}\) is a vector point function, then \(\nabla \cdot(f \vec{G})=\nabla f \cdot \vec{G}+f(\nabla \cdot \vec{G})\).

ii) If \(f\) is a scalar point function and \(\vec{G}\) is a vector point function, then \(\nabla \times(f \vec{G})=\nabla f \times \vec{G}+f(\nabla \times G)\).

iii) If \(\vec{F}\) and \(\vec{G}\) are vector point functions, then \(\nabla(\vec{F} \cdot \vec{G})=(\vec{F} \cdot \nabla) \vec{G}+(\vec{G} \cdot \nabla) \vec{F}+\vec{F} \times(\nabla \times \vec{G})+\vec{G} \times(\nabla \times \vec{F})\)

iv) If \(\vec{F}\) and \(\vec{G}\) are vector point functions then \(\begin{aligned} \nabla \cdot(\vec{F} \times \vec{G}) &=\vec{G} \cdot(\nabla \times \vec{F})-\vec{F} \cdot(\nabla \times \vec{G}) \\ \text { i.e., div } \vec{F} \times \vec{G} &=\vec{G} \cdot \operatorname{Curt} \vec{F}-\vec{F} \cdot \operatorname{Curt} \vec{G} \end{aligned}\)

v) If \(\vec{F}\) and \(\vec{G}\) are vector product functions, then \(\nabla \times(\vec{F} \times \vec{G})=\vec{F}(\nabla \cdot \vec{G})-\vec{G}(\nabla \cdot \vec{F})+(\vec{G} \cdot \nabla) \vec{F}-(\vec{F} \cdot \nabla) \vec{G}\)

\(\nabla\) operator applied twice to point functions.

i) If \(f\) is scalar point function, then div grad \(f=\nabla^{2} f=\dfrac{\partial^{2} f}{\partial x^{2}}+\dfrac{\partial^{2} f}{\partial y^{2}}+\dfrac{\partial^{2} f}{\partial z^{2}}\).

ii) If \(\vec{F}\) is a vector point function, then div \(\operatorname{curl} \vec{F}=0\).

iii) If \(\vec{F}\) is a vector point function, then \(\operatorname{curl}(\operatorname{Cur} \vec{F})=\nabla \times(\nabla \times \vec{F})=\nabla(\nabla \cdot \vec{F})-\nabla^{2} \vec{F}\).

PYQs

Vector Identities

1) Let \(\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}\). Show that \(curl(curl \vec{v})=grad(div \vec{v})-\nabla^2 \vec{v}\)

[2018, 12M]

2) Calculate \(\nabla^{2}\left(r^{n}\right)\) and find its expression in terms of \(r\) and \(n\), \(r\) being the distance of any point \((x, y, z)\) from the origin, \(n\) being a constant and \(\nabla^{2}\) being the Laplace operator.

[2013, 10M]

3) If \(u\) and \(v\) are two scalar fields and \(\vec{f}\) is a vector field, such that \(u \vec{f}=grad v\), find the value of \(\vec{f} \cdot curl \vec{f}\).

[2011, 10M]

4) Prove that \(\operatorname{div}(f \vec{V})=f(d i v \overline{V})+(\operatorname{grad} . f) \vec{V}\), where \(f\) is a scalar function.

[2010, 20M]

5) Prove that \(\nabla^{2} f(r)=\dfrac{d^{2} f}{d r^{2}}+\dfrac{2}{r} \dfrac{d f}{d r}\) where \(r=\left(x^{2}+y^{2}+z^{2}\right)^{1/2}\). Hence find \(f(x)\) such that \(\nabla^{2} f(r)=0\).

[2008, 15M]

6) Show that \(\operatorname{div}\left(\operatorname{grad} r^{n}\right)=n(n+1) r^{n-2}\), where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\).

[2009, 12M]

7) Show that curl \(\left(k \times \operatorname{grad} \dfrac{1}{r}\right)+\operatorname{grad}\left(k \operatorname{grad} \dfrac{1}{r}\right)=0\) where \(r\) is the distance from the origin and \(k\) is the unit vector in the direction \(OZ\).

[2005, 15M]

8) Prove the identity \(\nabla(\overline{A} \cdot \overline{B})=(\overline{B} . \overline{A}) \overline{A}+(\overline{A} \cdot \nabla) \overline{B}+\overline{B} \times(\nabla \times \overline{A})+\overline{A} \times(\nabla \times \overline{B})\).

[2004, 15M]

9) Derive the identity \(\iiint_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\iint_{S}(\phi \nabla \psi-\psi \nabla \phi) \hat{n} d S\) where \(V\) is the volume bounded by the closed surface \(S\).

[2004, 15M]

10) Show that if \(a^{\prime}\), \(b^{\prime}\) and \(c^{\prime}\) are the reciprocals of the non-coplanar vectors \(a\), \(b\) and \(c\), then any vector \(r\) may be expressed as \(r=\left(r . a^{\prime}\right) a+\left(r . c^{\prime}\right) c\).

[2003, 12M]

11) Prove the identity \(\nabla A^{2}=2(A . \nabla) A+2 A \times(\nabla \times A)\) where \(\nabla=\hat{i} \dfrac{\partial}{\partial x}+\hat{j} \dfrac{\partial}{\partial y}+\hat{k} \dfrac{\partial}{\partial z}\).

[2003, 15M]

12) Let \(\overline{R}\) be the unit vector along the vector \(\overline{r}(t)\). Show that \(\overline{R} \times \dfrac{\overline{d R}}{d t}=\dfrac{\overline{r}}{r^{2}} \times \dfrac{\overline{d r}}{d t}\) where \(r= \vert r \vert\).

[2002, 15M]

13) Show that \((curl \overline{v})=\operatorname{grad}(d i v \overline{v})-\nabla^{2} v\).

[2002, 15M]

14) Show that \(curl \dfrac{a \times r}{r^{3}}=-\dfrac{a}{r^{3}}+\dfrac{3 r}{r^{5}}(a \cdot r)\), where \(a\) is constant vector.

[2001, 12M]

Vector Equations

1) Prove that the vector \(\vec{a}=3 \vec{i}+\hat{j}-2 \hat{k}\), \(\vec{b}=-\hat{i}+3 \hat{j}+4 \hat{k}\), \(\vec{c}=4 \hat{i}-2 \hat{j}-6 \hat{k}\) can from the sides of a triangle. Find the length of the medians of the triangle.

[2016, 10M]

2) If \(\vec{A}=x^{2} y z \vec{i}-2 x z^{3} \vec{j}+x z^{2} \vec{k}\), \(\vec{B}=2 z \vec{i}+y \vec{j}-x^{2} \vec{k}\), find the value of \(\dfrac{\partial^{2}}{\partial x \partial y}(\vec{A}+\vec{B})\) at \((1,0,-2)\).

[2012, 12M]

3) If \(\overline{\mathrm{A}}=2 \hat{i}+\hat{k}\), \(\overline{\mathrm{B}}=\hat{i}+\hat{j}+\hat{k}\), \(\overline{C}=4 \hat{i}-3 \hat{j}-7 \hat{k}\), determine a vector \(\overline{R}\) satisfying the vector equation \(\overline{R} \times \overline{B}=\overline{C} \times \overline{B}\) and \(\overline{R} \cdot \overline{\mathrm{A}}=0\).

[2006, 15M]

4) Let \(D\) be a closed and bounded region having boundary \(S\). Further, let \(f\) is a scalar function having second partial derivatives defined on it. Show that \(\iint_{s}(\text { fgradf }) \hat{n} d s=\iiint_{v}\left[\vert grad f \vert^{2}+f \nabla^{2} f\right] d v\). Hence \(\iint_{s}f(\text { grad }f) \cdot \hat{n} d s\) or otherwise evaluate for \(f=2 x+y+2 z\) over \(s: x^{2}+y^{2}+z^{2}=4\).

[2002, 15M]

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