2.10: Aşagida verilen vektorler icin overrightarrow (nabla )(overrightarrow (A)cdot overrightarrow (B)) yi hesaplayiniz. overrightarrow (A)=rsinphi hat (f)+(cosTheta )/(r^2)hat (Theta )+sin^2phi hat (phi ), overrightarrow (B)=rhat (f)+rcosTheta hat (Theta )

Verilen vektorlar:<br /><br />\[<br />\overrightarrow{A} = r \sin \phi \hat{f} + \frac{\cos \Theta}{r^2} \hat{\Theta} + \sin^2 \phi \hat{\phi}<br />\]<br /><br />\[<br />\overrightarrow{B} = r \hat{f} + r \cos \Theta \hat{\Theta}<br />\]<br /><br />Bu vektorların iç çarpımını hesaplayalım:<br /><br />\[<br />\overrightarrow{A} \cdot \overrightarrow{B} = \left( r \sin \phi \right) \cdot r + \left( \frac{\cos \Theta}{r^2} \right) \cdot r \cos \Theta + \left( \sin^2 \phi \right) \cdot 0<br />\]<br /><br />\[<br />= r^2 \sin \phi + \frac{r \cos \Theta \cos \Theta}{r^2}<br />\]<br /><br />\[<br />= r^2 \sin \phi + \frac{\cos^2 \Theta}{r}<br />\]<br /><br />Şimdi, bu iç çarpımın gradientini hesaplayalım:<br /><br />\[<br />\nabla (\overrightarrow{A} \cdot \overrightarrow{B}) = \nabla \left( r^2 \sin \phi + \frac{\cos^2 \Theta}{r} \right)<br />\]<br /><br />Bu ifadeyi bileşenlerine göre türetelim:<br /><br />\[<br />= \nabla (r^2 \sin \phi) + \nabla \left( \frac{\cos^2 \Theta}{r} \right)<br />\]<br /><br />\[<br />= \nabla (r^2) \cdot \sin \phi + r^2 \cdot \nabla (\sin \phi) + \nabla \left( \frac{\cos^2 \Theta}{r} \right)<br />\]<br /><br />\[<br />= 2r \sin \phi \hat{r} + r^2 \cos \phi \hat{\phi} - \frac{2 \cos \Theta \sin \Theta}{r^2} \hat{r}<br />\]<br /><br />Sonuç olarak:<br /><br />\[<br />\nabla (\overrightarrow{A} \cdot \overrightarrow{B}) = \left( 2r \sin \phi - \frac{2 \cos \Theta \sin \Theta}{r^2} \right) \hat{r} + r^2 \cos \phi \hat{\phi}<br />\]