= Solution
Write $T=\dot\gamma$ along an affinely parametrized <geodesic> and $D_t=\nabla_T$ for the <covariant derivative> from the <Levi-Civita connection>. We use the <Riemann curvature tensor> convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$, for which positive <sectional curvature> means $\langle R(X,Y)Y,X\rangle>0$ on an orthonormal pair. A <Jacobi field> is a smooth <vector field> along $\gamma$ satisfying
$$
\boxed{D_t^2J+R(J,T)T=0.}
$$
Such a field describes the first-order separation of nearby <geodesics>, as the <geodesic variation> construction in part (c) shows.
For $t_0>0$, $\gamma(t_0)$ is a <conjugate point> of $\gamma(0)$ along this <geodesic> if a nonzero <Jacobi field> satisfies $J(0)=J(t_0)=0$. The <multiplicity of a conjugate point> is
$$
\boxed{\dim\{J:D_t^2J+R(J,T)T=0,\ J(0)=J(t_0)=0\}.}
$$
It is equivalently the dimension of the <kernel> of the <differential> of the <Riemannian exponential map> at $t_0\dot\gamma(0)$, by <Jacobi fields vanishing at their initial point>. Conjugacy concerns the specified <geodesic> and its two parameter values, even when its endpoints represent the same point of the <Riemannian manifold>.
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